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<hr />
<div>== Introduction ==<br />
Humans have surprisingly robust language processing systems which can easily overcome typos, e.g.<br />
<br />
* "Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist and lsat ltteer be at the rghit pclae."<br />
<br />
A person's ability to read this text comes as no surprise to the Psychology literature<br />
# Saberi & Perrott (1999) found that this robustness extends to audio as well.<br />
# Rayner et al. (2006) found that in noisier settings reading comprehension only slowed by 11%.<br />
# McCusker et al. (1981) found that the common case of swapping letters could often go unnoticed by the reader.<br />
# Mayall et al (1997) shows that we rely on word shape.<br />
# Reicher, 1969; Pelli et al., (2003) found that we can switch between whole word recognition but the first and last letter positions are required to stay constant for comprehension<br />
<br />
However, neural machine translation (NMT) systems are brittle. i.e. The Arabic word<br />
[[File:Good_morning.PNG]] means a blessing for good morning, however [[File:Hunt.PNG]] means hunt or slaughter. <br />
<br />
Facebook's MT system mistakenly confused two words that only differ by one character, a situation that is challenging for a character-based NMT system.<br />
<br />
The figure below shows the performance translating German to English as a function of the percent of German words modified. Here two types of noise are shown: (1) In blue, random permutation of the word and (2) In green, swapping a pair of adjacent letters that does not include the first or last letter of the word. The important thing to note is that even small amounts of noise lead to substantial drops in performance.<br />
<br />
[[File:BLEU_plot.PNG]] <br />
<br />
BLEU (bilingual evaluation understudy) is an algorithm for evaluating the quality of text which has been machine-translated from one natural language to another. Quality is considered to be the correspondence between a machine's output and that of a human: "the closer a machine translation is to a professional human translation, the better it is". BLEU is between 0 and 1.<br />
<br />
This paper explores two simple strategies for increasing model robustness:<br />
# using structure-invariant representations (character CNN representation)<br />
# robust training on noisy data, a form of adversarial training.<br />
<br />
The goal of the paper is two-fold:<br />
# to initiate a conversation on robust training and modeling techniques in NMT<br />
# to promote the creation of better and more linguistically accurate artificial noise to be applied to new languages and tasks<br />
<br />
== Adversarial examples ==<br />
The growing literature on adversarial examples has demonstrated how dangerous it can be to have brittle machine learning systems being used so pervasively in the real world. Small changes to the input can lead to dramatic<br />
failures of deep learning models. This leads to a potential for malicious attacks using adversarial examples. An important distinction is often drawn between white-box attacks, where adversarial examples are generated with<br />
access to the model parameters, and black-box attacks, where examples are generated without such access.<br />
<br />
The paper devises simple methods for generating adversarial examples for NMT. They do not assume any access to the NMT models' gradients, instead relying on cognitively-informed and naturally occurring language errors to generate noise.<br />
<br />
== MT system ==<br />
The authors experiment with three different NMT systems with access to character information at different levels.<br />
# Use <code>char2char</code>, the fully character-level model of (Lee et al. 2017). This model processes a sentence as a sequence of characters. The encoder works as follows: the characters are embedded as vectors, and then the sequence of vectors is fed to a convolutional layer. The sequence output by the convolutional layer is then shortened by max pooling in the time dimension. The output of the max-pooling layer is then fed to a four-layer highway network (Srivasta et al. 2015), and the output of the highway network is in turn fed to a bidirectional GRU, producing a sequence of hidden units. The sequence of hidden units is then processed by the decoder, a GRU with attention, to produce probabilities over sequences of output characters.<br />
# Use <code>Nematus</code> (Sennrich et al., 2017), a popular NMT toolkit. It is another sequence-to-sequence model with several architecture modifications, especially operating on sub-word units using byte-pair encoding. Byte-pair encoding (Sennich et al. 2015, Gage 1994) is an algorithm according to which we begin with a list of characters as our symbols, and repeatedly fuse common combinations to create new symbols. For example, if we begin with the letters a to z as our symbol list, and we find that "th" is the most common two-letter combination in a corpus, then we would add "th" to our symbol list in the first iteration. After we have used this algorithm to create a symbol list of the desired size, we apply a standard encoder-decoder with attention.<br />
# Use an attentional sequence-to-sequence model with a word representation based on a character convolutional neural network (<code>charCNN</code>). The <code>charCNN</code> model is similar to <code>char2char</code>, but uses a shallower highway network and, although it reads the input sentence as characters, it produces as output a probability distribution over words, not characters.<br />
<br />
== Data ==<br />
=== MT Data ===<br />
The authors use the TED talks parallel corpus prepared for IWSLT 2016 (Cettolo et al., 2012) for testing all of the NMT systems.<br />
<br />
[[File:Table1x.PNG]]<br />
<br />
=== Natural and Artificial Noise ===<br />
==== Natural Noise ====<br />
The three languages, French, German, and Czech, each have their own frequent natural errors. The corpora of edits used for these languages are:<br />
<br />
# French : Wikipedia Correction and Paraphrase Corpus (WiCoPaCo)<br />
# German : RWSE Wikipedia Correction Dataset and The MERLIN corpus<br />
# Czech : CzeSL Grammatical Error Correction Dataset (CzeSL-GEC) which is a manually annotated dataset of essays written by both non-native learners of Czech and Czech pupils<br />
<br />
The authors harvested naturally occurring errors (typos, misspellings, etc.) corresponding to these three languages from available corpora of edits to build a look-up table of possible lexical replacements.<br />
<br />
They insert these errors into the source-side of the parallel data by replacing every word in the corpus with an error if one exists in our dataset. When there is more than one possible replacement to choose, words for which there is no error, are sampled uniformly and kept as is.<br />
<br />
==== Synthetic Noise ====<br />
In addition to naturally collected sources of error, the authors also experiment with four types of synthetic noise: Swap, Middle Random, Fully Random, and Key Typo. <br />
# <code>Swap</code>: The first and simplest source of noise is swapping two letters (do not alter the first or last letters, only apply to words of length >=4).<br />
# <code>Middle Random</code>: Randomize the order of all the letters in a word except for the first and last (only apply to words of length >=4).<br />
# <code>Fully Random</code> Completely randomized words.<br />
# <code>Keyboard Typo</code> Randomly replace one letter in each word with an adjacent key<br />
<br />
[[File:Table3x.PNG]]<br />
<br />
Table 3 shows BLEU scores of models trained on clean (Vanilla) texts and tested on clean and noisy<br />
texts. All models suffer a significant drop in BLEU when evaluated on noisy texts. This is true<br />
for both natural noise and all kinds of synthetic noise. The more noise in the text, the worse the<br />
translation quality, with random scrambling producing the lowest BLEU scores.<br />
<br />
In contrast to the poor performance of these methods in the presence of noise, humans can perform very well as mentioned in the introduction. The table below shows the translations performed by a German native-speaker human, not familiar with the meme and three machine translation methods. Clearly, the machine translation methods failed. <br />
<br />
[[File:paper16_tab4.png]]<br />
<br />
The author also examined improvements by using a simple spell checker. The author tried correcting error through Google's spell checker by simply accepting the first suggestion on the detected mistake. There was a small improvement in French and German translations, and a small drop in accuracy for the Czech translation due to more complex grammar. The author concluded using existing spell checkers would not improve the accuracy to be comparable with vanilla text. The results are shown in the table below.<br />
<br />
[[File:paper16_tab5.png]]<br />
<br />
== Dealing with noise ==<br />
=== Structure Invariant Representations ===<br />
The three NMT models are all sensitive to word structure. The <code>char2char</code> and <code>charCNN</code> models both have convolutional layers on character sequences, designed to capture character n-grams (which are sequences of characters or words, of length n). The model in <code>Nematus</code> is based on sub-word units obtained with byte pair encoding (where common consecutive characters are replaced with a unique byte that does not occur in the data). It thus relies on character order.<br />
<br />
The simplest way to improve such a model is to take the average character embeddings as a word representation. This model, referred to as <code>meanChar</code>, first generates a word representation by averaging character embeddings, and then proceeds with a word-level encoder similar to the <code>charCNN</code> model.<br />
<br />
[[File:Table5x.PNG]]<br />
<br />
<code>meanChar</code> is good with the other three scrambling errors (Swap, Middle Random and Fully Random), but bad with Keyboard errors and Natural errors.<br />
<br />
=== Black-Box Adversarial Training ===<br />
<br />
<code>charCNN</code> Performance<br />
[[File:Table6x.PNG]]<br />
<br />
Here is the result of the translation of the scrambled meme:<br />
“According to a study of Cambridge University, it doesn’t matter which technology in a word is going to get the letters in a word that is the only important thing for the first and last letter.”<br />
<br />
== Analysis ==<br />
=== Learning Multiple Kinds of Noise in <code>charCNN</code> ===<br />
<br />
As Table 6 above shows, <code>charCNN</code> models performed quite well across different noise types on the test set when they are trained on a mix of noise types, which led the authors to speculate that filters from different convolutional layers learned to be robust to different types of noises. To test this hypothesis, they analyzed the weights learned by <code>charCNN</code> models trained on two kinds of input: completely scrambled words (Rand) without other kinds of noise, and a mix of Rand+Key+Nat kinds of noise. For each model, they computed the variance across the filter dimension for each one of the 1000 filters and for each one of the 25 character embedding dimensions, which were then averaged across the filters to yield 25 variances. <br />
<br />
As Figure 2 below shows, the variances for the ensemble model are higher and more varied, which indicates that the filters learned different patterns and the model differentiated between different character embedding dimensions. Under the random scrambling scheme, there should be no patterns for the model to learn, so it makes sense for the filter weights to stay close uniform weights, hence the consistently lower variance measures.<br />
<br />
[[File:Table7x.PNG]]<br />
<br />
=== Richness of Natural Noise ===<br />
<br />
The synthetic noise used in this paper appears to be very different from natural noise. This is evident because none of the modes trained only on synthetic noise demonstrated good performance on natural noise. Therefore, the authors say that the noise models used in this paper are not representative of real noise and that a more sophisticated model using explicit phonemic and linguistic knowledge is required if an error-free corpus is to be augmented with error for training.<br />
<br />
== Conclusion ==<br />
In this work, the authors have shown that character-based NMT models are extremely brittle and tend to break when presented with both natural and synthetic kinds of noise. After a comparison of the models, they found that a character-based CNN can learn to<br />
address multiple types of errors that are seen in training.<br />
For the future work, the author suggested generating more realistic synthetic noise by using phonetic and syntactic structure. Also, they suggested that a better NMT architecture could be designed which can be robust to noise without seeing it in the training data.<br />
<br />
== Criticism ==<br />
A major critique of this paper is that the solutions presented do not adequately solve the problem. The response to the meanChar architecture has been mostly negative and the method of noise injection has been seen as a simple start. However, the authors have acknowledged these critiques stating that they realize their solution is just a starting point. They argue that this paper has opened the discussion on dealing with noise in machine translation which has been mostly left untouched. Also these solutions/models still do not tackle the problem of natural noise as the models trained on the synthetic noise don't generalize well to natural noise.<br />
<br />
== References ==<br />
# Yonatan Belinkov and Yonatan Bisk. Synthetic and Natural Noise Both Break Neural Machine Translation. In ''International Conference on Learning Representations (ICLR)'', 2017.<br />
# Mauro Cettolo, Christian Girardi, and Marcello Federico. WIT: Web Inventory of Transcribed and Translated Talks. In ''Proceedings of the 16th Conference of the European Association for Machine Translation (EAMT)'', pp. 261–268, Trento, Italy, May 2012.<br />
# Jason Lee, Kyunghyun Cho, and Thomas Hofmann. Fully Character-Level Neural Machine Translation without Explicit Segmentation. ''Transactions of the Association for Computational Linguistics (TACL)'', 2017.<br />
# Rico Sennrich, Orhan Firat, Kyunghyun Cho, Alexandra Birch, Barry Haddow, Julian Hitschler, Marcin Junczys-Dowmunt, Samuel Laubli, Antonio Valerio Miceli Barone, Jozef Mokry, and Maria Nadejde. Nematus: a Toolkit for Neural Machine Translation. In ''Proceedings of the Software Demonstrations of the 15th Conference of the European Chapter of the Association for Computational Linguistics'', pp. 65–68, Valencia, Spain, April 2017. Association for Computational Linguistics. URL http://aclweb.org/anthology/E17-3017.<br />
# Aurlien Max and Guillaume Wisniewski. Mining Naturally-occurring Corrections and Paraphrases from Wikipedias Revision History. In Proceedings of the Seventh conference on International Language Resources and Evaluation (LREC’10), Valletta, Malta, may 2010. European Language Resources Association (ELRA). ISBN 2-9517408-6-7. URL https://wicopaco.limsi.fr.<br />
# Katrin Wisniewski, Karin Schne, Lionel Nicolas, Chiara Vettori, Adriane Boyd, Detmar Meurers, Andrea Abel, and Jirka Hana. MERLIN: An online trilingual learner corpus empirically grounding the European Reference Levels in authentic learner data, 10 2013. URL https://www.ukp.tu-darmstadt.de/data/spelling-correction/rwse-datasets.<br />
# Torsten Zesch. Measuring Contextual Fitness Using Error Contexts Extracted from the Wikipedia Revision History. In Proceedings of the 13th Conference of the European Chapter of the Association for Computational Linguistics, pp. 529–538, Avignon, France, April 2012. Association for Computational Linguistics.<br />
# Suranjana Samanta and Sameep Mehta. Towards Crafting Text Adversarial Samples. arXiv preprint arXiv:1707.02812, 2017. Karel Sebesta, Zuzanna Bedrichova, Katerina Sormov́a, Barbora Stindlov́a, Milan Hrdlicka, Tereza Hrdlickov́a, Jiŕı Hana, Vladiḿır Petkevic, Toḿas Jeĺınek, Svatava Skodov́a, Petr Janes, Katerina Lund́akov́a, Hana Skoumalov́a, Simon Sĺadek, Piotr Pierscieniak, Dagmar Toufarov́a, Milan Straka, Alexandr Rosen, Jakub Ńaplava, and Marie Poĺackova. CzeSL grammatical error correction dataset (CzeSL-GEC). Technical report, LINDAT/CLARIN digital library at the Institute of Formal and Applied Linguistics, Charles University, 2017. URL https://lindat.mff.cuni.cz/repository/xmlui/handle/11234/1-2143.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Multi-scale_Dense_Networks_for_Resource_Efficient_Image_Classification&diff=36061Multi-scale Dense Networks for Resource Efficient Image Classification2018-04-03T06:10:16Z<p>J3ku: Formatting, add example</p>
<hr />
<div>= Introduction = <br />
<br />
Multi-Scale Dense Networks, MSDNets, are designed to address the growing demand for efficient object recognition. The issue with existing recognition networks is that they are either efficient networks, but don't do well on hard examples, or large networks that do well on all examples but require a large amount of resources. For example, the winner of the COCO 2016 competition was an [http://image-net.org/challenges/talks/2016/GRMI-COCO-slidedeck.pdf ensemble of CNNs], which are likely far too resource-heavy to be used in any resource-limited application.<br />
<br />
In order to be efficient on all difficulties MSDNets propose a structure that can accurately output classifications for varying levels of computational requirements. The two cases that are used to evaluate the network are:<br />
* Anytime Prediction: What is the best prediction the network can provide when suddenly prompted?<br />
* Budget Batch Predictions: Given a maximum amount of computational resources, how well does the network do on the batch?<br />
<br />
= Related Networks =<br />
<br />
== Computationally Efficient Networks ==<br />
<br />
Much of the existing work on convolution networks that are computationally efficient at test time focus on reducing model size after training. Many existing methods for refining an accurate network to be more efficient include weight pruning [3,4,5], quantization of weights [6,7] (during or after training), and knowledge distillation [8,9], which trains smaller student networks to reproduce the output of a much larger teacher network. The proposed work differs from these approaches as it trains a single model which trades computation efficiency for accuracy at test time without re-training or finetuning.<br />
<br />
== Resource Efficient Networks == <br />
<br />
Unlike the above, resource efficient concepts consider limited resources as a part of the structure/loss.<br />
Examples of work in this area include: <br />
* Efficient variants to existing state of the art networks<br />
* Gradient boosted decision trees, which incorporate computational limitations into the training<br />
* Fractal nets<br />
* Adaptive computation time method<br />
<br />
== Related architectures ==<br />
<br />
MSDNets pull on concepts from a number of existing networks:<br />
* Neural fabrics and others, are used to quickly establish a low resolution feature map, which is integral for classification.<br />
* Deeply supervised nets, introduced the incorporation of multiple classifiers throughout the network<br />
* The feature concatenation method from DenseNets allows the later classifiers to not be disrupted by the weight updates from earlier classifiers.<br />
<br />
= Problem Setup =<br />
The authors consider two settings that impose computational constraints at prediction time.<br />
<br />
== Anytime Prediction ==<br />
In the anytime prediction setting (Grubb & Bagnell, 2012), there is a finite computational budget <math>B > 0</math> available for each test example <math>x</math>. Once the budget is exhausted, the prediction for the class is output using early exit. The budget is nondeterministic and varies per test instance.<br />
<br />
== Budgeted Batch Classification ==<br />
In the budgeted batch classification setting, the model needs to classify a set of examples <math>D_test = {x_1, . . . , x_M}</math> within a finite computational budget <math>B > 0</math> that is known in advance.<br />
<br />
= Multi-Scale Dense Networks =<br />
<br />
== Integral Contributions ==<br />
<br />
The way MSDNets aims to provide efficient classification with varying computational costs is to create one network that outputs results at depths. While this may seem trivial, as intermediate classifiers can be inserted into any existing network, two major problems arise.<br />
<br />
=== Coarse Level Features Needed For Classification ===<br />
<br />
[[File:paper29 fig3.png | 700px|thumb|center]]<br />
<br />
The term coarse level feature refers to a set of filters in a CNN with low resolution. There are several ways to create such features. These methods are typically refereed to as down sampling. Some example of layers that perform this function are: max pooling, average pooling and convolution with strides. In this architecture, convolution with strides will be used to create coarse features. <br />
<br />
Coarse level features are needed to gain context of scene. In typical CNN based networks, the features propagate from fine to coarse. Classifiers added to the early, fine featured, layers do not output accurate predictions due to the lack of context.<br />
<br />
Figure 3 depicts relative accuracies of the intermediate classifiers and shows that the accuracy of a classifier is highly correlated with its position in the network. It is easy to see, specifically with the case of ResNet, that the classifiers improve in a staircase pattern. All of the experiments were performed on Cifar-100 dataset and it can be seen that the intermediate classifiers perform worst than the final classifiers, thus highlighting the problem with the lack of coarse level features early on.<br />
<br />
To address this issue, MSDNets proposes an architecture in which uses multi scaled feature maps. The feature maps at a particular layer and scale are computed by concatenating results from up to two convolutions: a standard convolution is first applied to same-scale features from the previous layer to pass on high-resolution information that subsequent layers can use to construct better coarse features, and if possible, a strided convolution is also applied on the finer-scale feature map from the previous layer to produce coarser features amenable to classification. The network is quickly formed to contain a set number of scales ranging from fine to coarse. These scales are propagated throughout, so that for the length of the network there are always coarse level features for classification and fine features for learning more difficult representations.<br />
<br />
=== Training of Early Classifiers Interferes with Later Classifiers ===<br />
<br />
When training a network containing intermediate classifiers, the training of early classifiers will cause the early layers to focus on features for that classifier. These learned features may not be as useful to the later classifiers and degrade their accuracy.<br />
<br />
MSDNets use dense connectivity to avoid this issue. By concatenating all prior layers to learn future layers, the gradient propagation is spread throughout the available features. This allows later layers to not be reliant on any single prior, providing opportunities to learn new features that priors have ignored.<br />
<br />
== Architecture ==<br />
<br />
[[File:MSDNet_arch.png | 700px|thumb|center|Left: the MSDNet architecture. Right: example calculations for each output given 3 scales and 4 layers.]]<br />
<br />
The architecture of MSDNet is a structure of convolutions with a set number of layers and a set number of scales. Layers allow the network to build on the previous information to generate more accurate predictions, while the scales allow the network to maintain coarse level features throughout.<br />
<br />
The first layer is a special, mini-CNN-network, that quickly fills all required scales with features. The following layers are generated through the convolutions of the previous layers and scales.<br />
<br />
Each output at a given s scale is given by the convolution of all prior outputs of the same scale, and the strided-convolution of all prior outputs from the previous scale. <br />
<br />
The classifiers consists of two convolutional layers, an average pooling layer and a linear layer and are run on the concatenation of all of the coarsest outputs from the preceding layers.<br />
<br />
=== Loss Function ===<br />
<br />
The loss is calculated as a weighted sum of each classifier's logistic loss: <br />
<br />
<math>\frac{1}{|\mathcal{D}|} \sum_{x,y \in \mathcal{D}} \sum_{k}w_k L(f_k) </math><br />
<br />
Here <math>w_i</math> represents the weights and <math>L(f_k)</math> represents the logistic loss of each classifier. The weighted loss is taken as an average over a set of training samples. The weights can be determined from a budget of computational power, but results also show that setting all to 1 is also acceptable.<br />
<br />
=== Computational Limit Inclusion ===<br />
<br />
When running in a budgeted batch scenario, the network attempts to provide the best overall accuracy. To do this with a set limit on computational resources, it works to use less of the budget on easy detections in order to allow more time to be spent on hard ones. <br />
In order to facilitate this, the classifiers are designed to exit when the confidence of the classification exceeds a preset threshold. To determine the threshold for each classifier, <math>|D_{test}|\sum_{k}(q_k C_k) \leq B </math> must be true. Where <math>|D_{test}|</math> is the total number of test samples, <math>C_k</math> is the computational requirement to get an output from the <math>k</math>th classifier, and <math>q_k </math> is the probability that a sample exits at the <math>k</math>th classifier. Assuming that all classifiers have the same base probability, <math>q</math>, then <math>q_k</math> can be used to find the threshold.<br />
<br />
=== Network Reduction and Lazy Evaluation ===<br />
There are two ways to reduce the computational needs of MSDNets:<br />
<br />
# Reduce the size of the network by splitting it into <math>S</math> blocks along the depth dimension and keeping the <math>(S-i+1)</math> scales in the <math>i^{\text{th}}</math> block.Whenever a scale is removed, a transition layer merges the concatenated features using 1x1 convolution and feeds the fine grained features to coarser scales.<br />
# Remove unnecessary computations: Group the computation in "diagonal blocks"; this propagates the example along paths that are required for the evaluation of the next classifier.<br />
<br />
The strategy of minimizing unnecessary computations when the computational budget is over is known as the ''lazy evaluation''.<br />
<br />
= Experiments = <br />
<br />
When evaluating on CIFAR-10 and CIFAR-100 ensembles and multi-classifier versions of ResNets and DenseNets, as well as FractalNet are used to compare with MSDNet. <br />
<br />
When evaluating on ImageNet ensembles and individual versions of ResNets and DenseNets are compared with MSDNets.<br />
<br />
== Anytime Prediction ==<br />
<br />
In anytime prediction MSDNets are shown to have highly accurate with very little budget, and continue to remain above the alternate methods as the budget increases. The authors attributed this to the fact that MSDNets are able to produce low-resolution feature maps well-suited for classification after just a few layers, in contrast to the high-resolution feature maps in early layers of ResNets or DenseNets. Ensemble networks need to repeat computations of similar low-level features repeatedly when new models need to be evaluated, so their accuracy results do not increase as fast when computational budget increases. <br />
<br />
[[File:MSDNet_anytime.png | 700px|thumb|center|Accuracy of the anytime classification models.]] [[File:cifar10msdnet.png | 700px|thumb|center|CIFAR-10 results.]]<br />
<br />
== Budget Batch ==<br />
<br />
For budget batch 3 MSDNets are designed with classifiers set-up for varying ranges of budget constraints. On both dataset options the MSDNets exceed all alternate methods with a fraction of the budget required.<br />
<br />
[[File:MSDNet_budgetbatch.png | 700px|thumb|center|Accuracy of the budget batch classification models.]]<br />
<br />
The following figure shows examples of what was deemed "easy" and "hard" examples by the network. The top row contains images of either red wine or volcanos that were easily classified, thus exiting the network early and reducing required computations. The bottom row contains examples of "hard" images that were incorrectly classified by the first classifier but were correctly classified by the last layer.<br />
<br />
[[File:MSDNet_visualizingearlyclassifying.png | 700px|thumb|center|Examples of "hard"/"easy" classification]]<br />
<br />
= Ablation study =<br />
Additional experiments were performed to shed light on multi-scale feature maps, dense connectivity, and intermediate classifiers. This experiment started with an MSDNet with six intermediate classifiers and each of these components were removed, one at a time. To make our comparisons fair, the computational costs of the full networks were kept similar by adapting the network width. After removing all the three components, a VGG-like convolutional network is obtained. The classification accuracy of all classifiers is shown in the image below.<br />
<br />
[[File:Screenshot_from_2018-03-29_14-58-03.png]]<br />
<br />
= Critique = <br />
<br />
The problem formulation and scenario evaluation were very well formulated, and according to independent reviews, the results were reproducible. Where the paper could improve is on explaining how to implement the threshold; it isn't very well explained how the use of the validation set can be used to set the threshold value.<br />
<br />
= Implementation =<br />
The following repository provides the source code for the paper, written by the authors: https://github.com/gaohuang/MSDNet<br />
<br />
= Sources =<br />
# Huang, G., Chen, D., Li, T., Wu, F., Maaten, L., & Weinberger, K. Q. (n.d.). Multi-Scale Dense Networks for Resource Efficient Image Classification. ICLR 2018. doi:1703.09844 <br />
# Huang, G. (n.d.). Gaohuang/MSDNet. Retrieved March 25, 2018, from https://github.com/gaohuang/MSDNet<br />
# LeCun, Yann, John S. Denker, and Sara A. Solla. "Optimal brain damage." Advances in neural information processing systems. 1990.<br />
# Hassibi, Babak, David G. Stork, and Gregory J. Wolff. "Optimal brain surgeon and general network pruning." Neural Networks, 1993., IEEE International Conference on. IEEE, 1993.<br />
# Li, Hao, et al. "Pruning filters for efficient convnets." arXiv preprint arXiv:1608.08710 (2016).<br />
# Hubara, Itay, et al. "Binarized neural networks." Advances in neural information processing systems. 2016.<br />
# Rastegari, Mohammad, et al. "Xnor-net: Imagenet classification using binary convolutional neural networks." European Conference on Computer Vision. Springer, Cham, 2016.<br />
# Cristian Bucilua, Rich Caruana, and Alexandru Niculescu-Mizil. Model compression. In ACM SIGKDD, pp. 535–541. ACM, 2006.<br />
# Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. In NIPS Deep Learning Workshop, 2014.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Tensorized_LSTMs&diff=36060stat946w18/Tensorized LSTMs2018-04-03T06:01:01Z<p>J3ku: </p>
<hr />
<div>= Presented by =<br />
<br />
Chen, Weishi(Edward)<br />
<br />
= Introduction =<br />
<br />
Long Short-Term Memory (LSTM) is a popular approach to boosting the ability of Recurrent Neural Networks to store longer term temporal information. The capacity of an LSTM network can be increased by widening and adding layers (illustrations will be provided later). <br />
<br />
<br />
However, usually the LSTM model introduces additional parameters, while LSTM with additional layers and wider layers increases the time required for model training and evaluation. As an alternative, this paper <Wider and Deeper, Cheaper and Faster: Tensorized LSTMs for Sequence Learning> has proposed a model based on LSTM called the '''Tensorized LSTM''' in which the hidden states are represented by '''tensors''' and updated via a '''cross-layer convolution'''. <br />
<br />
* By increasing the tensor size, the network can be widened efficiently without additional parameters since the parameters are shared across different locations in the tensor<br />
* By delaying the output, the network can be deepened implicitly with little additional run-time since deep computations for each time step are merged into temporal computations of the sequence. <br />
<br />
<br />
Also, the paper presents experiments that were conducted on five challenging sequence learning tasks to show the potential of the proposed model.<br />
<br />
= A Quick Introduction to RNN and LSTM =<br />
<br />
We consider the time-series prediction task of producing a desired output <math>y_t</math> at each time-step t∈ {1, ..., T} given an observed input sequence <math>x_{1:t} = {x_1,x_2, ···, x_t}</math>, where <math>x_t∈R^R</math> and <math>y_t∈R^S</math> are vectors. RNN learns how to use a hidden state vector <math>h_t ∈ R^M</math> to encapsulate the relevant features of the entire input history x1:t (indicates all inputs from to initial time-step to final step before predication - illustration given below) up to time-step t.<br />
<br />
\begin{align}<br />
h_{t-1}^{cat} = [x_t, h_{t-1}] \hspace{2cm} (1)<br />
\end{align}<br />
<br />
Where <math>h_{t-1}^{cat} ∈R^{R+M}</math> is the concatenation of the current input <math>x_t</math> and the previous hidden state <math>h_{t−1}</math>, which expands the dimensionality of intermediate information.<br />
<br />
The update of the hidden state h_t is defined as:<br />
<br />
\begin{align}<br />
a_{t} =h_{t-1}^{cat} W^h + b^h \hspace{2cm} (2)<br />
\end{align}<br />
<br />
and<br />
<br />
\begin{align}<br />
h_t = \Phi(a_t) \hspace{2cm} (3)<br />
\end{align}<br />
<br />
<math>W^h∈R^(R+M)xM </math> guarantees each hidden status provided by the previous step is of dimension M. <math> a_t ∈R^M </math> is the hidden activation, and φ(·) is the element-wise hyperbolic tangent. Finally, the output <math> y_t </math> at time-step t is generated by:<br />
<br />
\begin{align}<br />
y_t = \varphi(h_{t}^{cat} W^y + b^y) \hspace{2cm} (4)<br />
\end{align}<br />
<br />
where <math>W^y∈R^{M×S}</math> and <math>b^y∈R^S</math>, and <math>\varphi(·)</math> can be any differentiable function. Note that the <math>\phi</math> is a non-linear, element-wise function which generates hidden output, while <math>\varphi</math> generates the final network output.<br />
<br />
[[File:StdRNN.png|650px|center||Figure 1: Recurrent Neural Network]]<br />
<br />
One shortfall of RNN is the problem of vanishing/exploding gradients. This shortfall is significant, especially when modeling long-range dependencies. One alternative is to instead use LSTM (Long Short-Term Memory), which alleviates these problems by employing several gates to selectively modulate the information flow across each neuron. Since LSTMs have been successfully used in sequence models, it is natural to consider them for accommodating more complex analytical needs.<br />
<br />
[[File:LSTM_Gated.png|650px|center||Figure 2: LSTM]]<br />
<br />
= Structural Measurement of Sequential Model =<br />
<br />
We can consider the capacity of a network consists of two components: the '''width''' (the amount of information handled in parallel) and the depth (the number of computation steps). <br />
<br />
A way to '''widen''' the LSTM is to increase the number of units in a hidden layer; however, the parameter number scales quadratically with the number of units. To deepen the LSTM, the popular Stacked LSTM (sLSTM) stacks multiple LSTM layers. The drawback of sLSTM, however, is that runtime is proportional to the number of layers and information from the input is potentially lost (due to gradient vanishing/explosion) as it propagates vertically through the layers. This paper introduced a way to both widen and deepen the LSTM whilst keeping the parameter number and runtime largely unchanged. In summary, we make the following contributions:<br />
<br />
'''(a)''' Tensorize RNN hidden state vectors into higher-dimensional tensors, to enable more flexible parameter sharing and can be widened more efficiently without additional parameters.<br />
<br />
'''(b)''' Based on (a), merge RNN deep computations into its temporal computations so that the network can be deepened with little additional runtime, resulting in a Tensorized RNN (tRNN).<br />
<br />
'''(c)''' We extend the tRNN to an LSTM, namely the Tensorized LSTM (tLSTM), which integrates a novel memory cell convolution to help to prevent the vanishing/exploding gradients.<br />
<br />
= Method =<br />
<br />
Go through the methodology.<br />
<br />
== Part 1: Tensorize RNN hidden State vectors ==<br />
<br />
'''Definition:''' Tensorization is defined as the transformation or mapping of lower-order data to higher-order data. For example, the low-order data can be a vector, and the tensorized result is a matrix, a third-order tensor or a higher-order tensor. The ‘low-order’ data can also be a matrix or a third-order tensor, for example. In the latter case, tensorization can take place along one or multiple modes.<br />
<br />
[[File:VecTsor.png|320px|center||Figure 3: Vector Third-order tensorization of a vector]]<br />
<br />
'''Optimization Methodology Part 1:''' It can be seen that in an RNN, the parameter number scales quadratically with the size of the hidden state. A popular way to limit the parameter number when widening the network is to organize parameters as higher-dimensional tensors which can be factorized into lower-rank sub-tensors that contain significantly fewer elements, which is is known as tensor factorization. <br />
<br />
'''Optimization Methodology Part 2:''' Another common way to reduce the parameter number is to share a small set of parameters across different locations in the hidden state, similar to Convolutional Neural Networks (CNNs).<br />
<br />
'''Effects:''' This '''widens''' the network since the hidden state vectors are in fact broadcast to interact with the tensorized parameters. <br />
<br />
<br />
<br />
We adopt parameter sharing to cutdown the parameter number for RNNs, since compared with factorization, it has the following advantages: <br />
<br />
(i) '''Scalability,''' the number of shared parameters can be set independent of the hidden state size<br />
<br />
(ii) '''Separability,''' the information flow can be carefully managed by controlling the receptive field, allowing one to shift RNN deep computations to the temporal domain<br />
<br />
<br />
<br />
We also explicitly tensorize the RNN hidden state vectors, since compared with vectors, tensors have a better: <br />
<br />
(i) '''Flexibility,''' one can specify which dimensions to share parameters and then can just increase the size of those dimensions without introducing additional parameters<br />
<br />
(ii) '''Efficiency,''' with higher-dimensional tensors, the network can be widened faster w.r.t. its depth when fixing the parameter number (explained later). <br />
<br />
<br />
'''Illustration:''' For ease of exposition, we first consider 2D tensors (matrices): we tensorize the hidden state <math>h_t∈R^{M}</math> to become <math>Ht∈R^{P×M}</math>, '''where P is the tensor size,''' and '''M the channel size'''. We locally-connect the first dimension of <math>H_t</math> (which is P - the tensor size) in order to share parameters, and fully-connect the second dimension of <math>H_t</math> (which is M - the channel size) to allow global interactions. This is analogous to the CNN which fully-connects one dimension (e.g., the RGB channel for input images) to globally fuse different feature planes. Also, if one compares <math>H_t</math> to the hidden state of a Stacked RNN (sRNN) (see Figure Blow). <br />
<br />
[[File:Screen_Shot_2018-03-26_at_11.28.37_AM.png|160px|center||Figure 4: Stacked RNN]]<br />
<br />
[[File:ind.png|60px|center||Figure 4: Stacked RNN]]<br />
<br />
Then P is akin to the number of stacked hidden layers (vertical length in the graph), and M the size of each hidden layer (each white node in the graph). We start to describe our model based on 2D tensors, and finally show how to strengthen the model with higher-dimensional tensors.<br />
<br />
== Part 2: Merging Deep Computations ==<br />
<br />
Since an RNN is already deep in its temporal direction, we can deepen an input-to-output computation by associating the input <math>x_t</math> with a (delayed) future output. In doing this, we need to ensure that the output <math>y_t</math> is separable, i.e., not influenced by any future input <math>x_{t^{'}}</math> <math>(t^{'}>t)</math>. Thus, we concatenate the projection of <math>x_t</math> to the top of the previous hidden state <math>H_{t−1}</math>, then gradually shift the input information down when the temporal computation proceeds, and finally generate <math>y_t</math> from the bottom of <math>H_{t+L−1}</math>, where L−1 is the number of delayed time-steps for computations of depth L. <br />
<br />
An example with L= 3 is shown in Figure.<br />
<br />
[[File:tRNN.png|160px|center||Figure 5: skewed sRNN]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: skewed sRNN]]<br />
<br />
<br />
This is in fact a skewed sRNN (or tRNN without feedback). However, the method does not need to change the network structure and also allows different kinds of interactions as long as the output is separable; for example, one can increase the local connections and '''use feedback''' (shown in figure below), which can be beneficial for sRNNs (or tRNN). <br />
<br />
[[File:tRNN_wF.png|160px|center||Figure 5: skewed sRNN with F]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: skewed sRNN with F]]<br />
<br />
'''In order to share parameters, we update <math>H_t</math> using a convolution with a learnable kernel.''' In this manner we increase the complexity of the input-to-output mapping (by delaying outputs) and limit parameter growth (by sharing transition parameters using convolutions).<br />
<br />
To examine the resulting model mathematically, let <math>H^{cat}_{t−1}∈R^{(P+1)×M}</math> be the concatenated hidden state, and <math>p∈Z_+</math> the location at a tensor. The channel vector <math>h^{cat}_{t−1, p }∈R^M</math> at location p of <math>H^{cat}_{t−1}</math> (the p-th channel of H) is defined as:<br />
<br />
\begin{align}<br />
h^{cat}_{t-1, p} = x_t W^x + b^x \hspace{1cm} if p = 1 \hspace{1cm} (5)<br />
\end{align}<br />
<br />
\begin{align}<br />
h^{cat}_{t-1, p} = h_{t-1, p-1} \hspace{1cm} if p > 1 \hspace{1cm} (6)<br />
\end{align}<br />
<br />
where <math>W^x ∈ R^{R×M}</math> and <math>b^x ∈ R^M</math> (recall the dimension of input x is R). Then, the update of tensor <math>H_t</math> is implemented via a convolution:<br />
<br />
\begin{align}<br />
A_t = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (7)<br />
\end{align}<br />
<br />
\begin{align}<br />
H_t = \Phi{A_t} \hspace{2cm} (8)<br />
\end{align}<br />
<br />
where <math>W^h∈R^{K×M^i×M^o}</math> is the kernel weight of size K, with <math>M^i =M</math> input channels and <math>M^o =M</math> output channels, <math>b^h ∈ R^{M^o}</math> is the kernel bias, <math>A_t ∈ R^{P×M^o}</math> is the hidden activation, and <math>\circledast</math> is the convolution operator. Since the kernel convolves across different hidden layers, we call it the cross-layer convolution. The kernel enables interaction, both bottom-up and top-down across layers. Finally, we generate <math>y_t</math> from the channel vector <math>h_{t+L−1,P}∈R^M</math> which is located at the bottom of <math>H_{t+L−1}</math>:<br />
<br />
\begin{align}<br />
y_t = \varphi(h_{t+L−1}, _PW^y + b^y) \hspace{2cm} (9)<br />
\end{align}<br />
<br />
Where <math>W^y ∈R^{M×S}</math> and <math>b^y ∈R^S</math>. To guarantee that the receptive field of <math>y_t</math> only covers the current and previous inputs x1:t. (Check the Skewed sRNN again below):<br />
<br />
[[File:tRNN_wF.png|160px|center||Figure 5: skewed sRNN with F]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: skewed sRNN with F]]<br />
<br />
=== Quick Summary of Set of Parameters ===<br />
<br />
'''1. <math> W^x</math> and <math>b_x</math>''' connect input to the first hidden node<br />
<br />
'''2. <math> W^h</math> and <math>b_h</math>''' convolute between layers<br />
<br />
'''3. <math> W^y</math> and <math>b_y</math>''' produce output of each stages<br />
<br />
<br />
== Part 3: Extending to LSTMs==<br />
<br />
Similar to standard RNN, to allow the tRNN (skewed sRNN) to capture long-range temporal dependencies, one can straightforwardly extend it<br />
to a tLSTM by replacing the tRNN tensors:<br />
<br />
\begin{align}<br />
[A^g_t, A^i_t, A^f_t, A^o_t] = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (10)<br />
\end{align}<br />
<br />
\begin{align}<br />
[G_t, I_t, F_t, O_t]= [\Phi{(A^g_t)}, σ(A^i_t), σ(A^f_t), σ(A^o_t)] \hspace{2cm} (11)<br />
\end{align}<br />
<br />
Which are pretty similar to tRNN case, the main differences can be observes for memory cells of tLSTM (Ct):<br />
<br />
\begin{align}<br />
C_t= G_t \odot I_t + C_{t-1} \odot F_t \hspace{2cm} (12)<br />
\end{align}<br />
<br />
\begin{align}<br />
H_t= \Phi{(C_t )} \odot O_t \hspace{2cm} (13)<br />
\end{align}<br />
<br />
Note that since the previous memory cell <math>C_{t-1}</math> is only gated along the temporal direction, increasing the tensor size ''P'' might result in the loss of long-range dependencies from the input to the output.<br />
<br />
Summary of the terms: <br />
<br />
1. '''<math>\{W^h, b^h \}</math>:''' Kernel of size K <br />
<br />
2. '''<math>A^g_t, A^i_t, A^f_t, A^o_t \in \mathbb{R}^{P\times M}</math>:''' Activations for the new content <math>G_t</math><br />
<br />
3. '''<math>I_t</math>:''' Input gate<br />
<br />
4. '''<math>F_t</math>:''' Forget gate<br />
<br />
5. '''<math>O_t</math>:''' Output gate<br />
<br />
6. '''<math>C_t \in \mathbb{R}^{P\times M}</math>:''' Memory cell<br />
<br />
Then, see graph below for illustration:<br />
<br />
[[File:tLSTM_wo_MC.png |160px|center||Figure 5: tLSTM wo MC]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: tLSTM wo MC]]<br />
<br />
To further evolve tLSTM, we invoke the '''Memory Cell Convolution''' to capture long-range dependencies from multiple directions, we additionally introduce a novel memory cell convolution, by which the memory cells can have a larger receptive field (figure provided below). <br />
<br />
[[File:tLSTM_w_MC.png |160px|center||Figure 5: tLSTM w MC]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: tLSTM w MC]]<br />
<br />
One can also dynamically generate this convolution kernel so that it is both time - and location-dependent, allowing for flexible control over long-range dependencies from different directions. Mathematically, it can be represented in with the following formulas:<br />
<br />
\begin{align}<br />
[A^g_t, A^i_t, A^f_t, A^o_t, A^q_t] = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (14)<br />
\end{align}<br />
<br />
\begin{align}<br />
[G_t, I_t, F_t, O_t, Q_t]= [\Phi{(A^g_t)}, σ(A^i_t), σ(A^f_t), σ(A^o_t), ς(A^q_t)] \hspace{2cm} (15)<br />
\end{align}<br />
<br />
\begin{align}<br />
W_t^c(p) = reshape(q_{t,p}, [K, 1, 1]) \hspace{2cm} (16)<br />
\end{align}<br />
<br />
\begin{align}<br />
C_{t-1}^{conv}= C_{t-1} \circledast W_t^c(p) \hspace{2cm} (17)<br />
\end{align}<br />
<br />
\begin{align}<br />
C_t= G_t \odot I_t + C_{t-1}^{conv} \odot F_t \hspace{2cm} (18)<br />
\end{align}<br />
<br />
\begin{align}<br />
H_t= \Phi{(C_t )} \odot O_t \hspace{2cm} (19)<br />
\end{align}<br />
<br />
where the kernel <math>{W^h, b^h}</math> has additional <K> output channels to generate the activation <math>A^q_t ∈ R^{P×<K>}</math> for the dynamic kernel bank <math>Q_t∈R^{P × <K>}</math>, <math>q_{t,p}∈R^{<K>}</math> is the vectorized adaptive kernel at the location p of <math>Q_t</math>, and <math>W^c_t(p) ∈ R^{K×1×1}</math> is the dynamic kernel of size K with a single input/output channel, which is reshaped from <math>q_{t,p}</math>. Each channel of the previous memory cell <math>C_{t-1}</math> is convolved with <math>W_t^c(p)</math> whose values vary with <math>p</math>, to form a memory cell convolution, which produces a convolved memory cell <math>C_{t-1}^{conv} \in \mathbb{R}^{P\times M}</math>. Note the paper also employed a softmax function ς(·) to normalize the channel dimension of <math>Q_t</math>. which can also stabilize the value of memory cells and help to prevent the vanishing/exploding gradients. An illustration is provided below to better illustrate the process:<br />
<br />
[[File:MCC.png |240px|center||Figure 5: MCC]]<br />
<br />
To improve training, the authors introduced a new normalization technique for ''t''LSTM termed channel normalization (adapted from layer normalization), in which the channel vector are normalized at different locations with their own statistics. Note that layer normalization does not work well with ''t''LSTM, because lower level information is near the input and higher level information is near the output. Channel normalization (CN) is defined as: <br />
<br />
\begin{align}<br />
\mathrm{CN}(\mathbf{Z}; \mathbf{\Gamma}, \mathbf{B}) = \mathbf{\hat{Z}} \odot \mathbf{\Gamma} + \mathbf{B} \hspace{2cm} (20)<br />
\end{align}<br />
<br />
where <math>\mathbf{Z}</math>, <math>\mathbf{\hat{Z}}</math>, <math>\mathbf{\Gamma}</math>, <math>\mathbf{B} \in \mathbb{R}^{P \times M^z}</math> are the original tensor, normalized tensor, gain parameter and bias parameter. The <math>m^z</math>-th channel of <math>\mathbf{Z}</math> is normalized element-wisely: <br />
<br />
\begin{align}<br />
\hat{z_{m^z}} = (z_{m^z} - z^\mu)/z^{\sigma} \hspace{2cm} (21)<br />
\end{align}<br />
<br />
where <math>z^{\mu}</math>, <math>z^{\sigma} \in \mathbb{R}^P</math> are the mean and standard deviation along the channel dimension of <math>\mathbf{Z}</math>, and <math>\hat{z_{m^z}} \in \mathbb{R}^P</math> is the <math>m^z</math>-th channel <math>\mathbf{\hat{Z}}</math>. Channel normalization introduces very few additional parameters compared to the number of other parameters in the model.<br />
<br />
= Results and Evaluation =<br />
<br />
Summary of list of models tLSTM family (may be useful later):<br />
<br />
(a) sLSTM (baseline): the implementation of sLSTM with parameters shared across all layers.<br />
<br />
(b) 2D tLSTM: the standard 2D tLSTM.<br />
<br />
(c) 2D tLSTM–M: removing memory (M) cell convolutions from (b).<br />
<br />
(d) 2D tLSTM–F: removing (–) feedback (F) connections from (b).<br />
<br />
(e) 3D tLSTM: tensorizing (b) into 3D tLSTM.<br />
<br />
(f) 3D tLSTM+LN: applying (+) Layer Normalization.<br />
<br />
(g) 3D tLSTM+CN: applying (+) Channel Normalization.<br />
<br />
=== Efficiency Analysis ===<br />
<br />
'''Fundaments:''' For each configuration, fix the parameter number and increase the tensor size to see if the performance of tLSTM can be boosted without increasing the parameter number. Can also investigate how the runtime is affected by the depth, where the runtime is measured by the average GPU milliseconds spent by a forward and backward pass over one timestep of a single example. <br />
<br />
'''Dataset:''' The Hutter Prize Wikipedia dataset consists of 100 million characters taken from 205 different characters including alphabets, XML markups and special symbols. We model the dataset at the character-level, and try to predict the next character of the input sequence.<br />
<br />
All configurations are evaluated with depths L = 1, 2, 3, 4. Bits-per-character(BPC) is used to measure the model performance and the results are shown in the figure below.<br />
[[File:wiki.png |280px|center||Figure 5: WifiPerf]]<br />
[[File:Wiki_Performance.png |480px|center||Figure 5: WifiPerf]]<br />
<br />
=== Accuracy Analysis ===<br />
<br />
The MNIST dataset [35] consists of 50000/10000/10000 handwritten digit images of size 28×28 for training/validation/test. Two tasks are used for evaluation on this dataset:<br />
<br />
(a) '''Sequential MNIST:''' The goal is to classify the digit after sequentially reading the pixels in a scan-line order. It is therefore a 784 time-step sequence learning task where a single output is produced at the last time-step; the task requires very long range dependencies in the sequence.<br />
<br />
(b) '''Sequential Permuted MNIST:''' We permute the original image pixels in a fixed random order, resulting in a permuted MNIST (pMNIST) problem that has even longer range dependencies across pixels and is harder.<br />
<br />
In both tasks, all configurations are evaluated with M = 100 and L= 1, 3, 5. The model performance is measured by the classification accuracy and results are shown in the figure below.<br />
<br />
[[File:MNISTperf.png |480px|center]]<br />
<br />
<br />
<br />
[[File:Acc_res.png |480px|center||Figure 5: MNIST]]<br />
<br />
[[File:33_mnist.PNG|center|thumb|800px| This figure displays a visualization of the means of the diagonal channels of the tLSTM memory cells per task. The columns indicate the time steps and the rows indicate the diagonal locations. The values are normalized between 0 and 1.]]<br />
<br />
= Conclusions =<br />
<br />
The paper introduced the Tensorized LSTM, which employs tensors to share parameters and utilizes the temporal computation to perform the deep computation for sequential tasks. Then validated the model<br />
on a variety of tasks, showing its potential over other popular approaches.<br />
<br />
= Critique(to be edited) =<br />
<br />
= References =<br />
#Zhen He, Shaobing Gao, Liang Xiao, Daxue Liu, Hangen He, and David Barber. <Wider and Deeper, Cheaper and Faster: Tensorized LSTMs for Sequence Learning> (2017)<br />
#Ali Ghodsi, <Deep Learning: STAT 946 - Winter 2018></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Training_And_Inference_with_Integers_in_Deep_Neural_Networks&diff=36059Training And Inference with Integers in Deep Neural Networks2018-04-03T05:51:38Z<p>J3ku: Introduction</p>
<hr />
<div>== Introduction ==<br />
<br />
Deep neural networks have enjoyed much success in all manners of tasks, but it is common for these networks to be complicated and have high memory requirements while performing many floating-point operations (FLOPs). As a result, running many of these models are very expensive in terms of energy use, and using state-of-the-art networks in applications where energy is limited can be very difficult. In order to overcome this and allow use of these networks in situations with low energy availability, the energy costs must be reduced while trying to maintain as high network performance as possible and/or practical.<br />
<br />
Most existing methods focus on reducing the energy requirements during inference rather than training. Since training with SGD requires accumulation, training usually has higher precision demand than inference. Most of the existing methods focus on how to compress a model for inference, rather than during training. This paper proposes a framework to reduce complexity both during training and inference through the use of integers instead of floats. The authors address how to quantize all operations and operands as well as examining the bitwidth requirement for SGD computation & accumulation. Using integers instead of floats results in energy-savings because integer operations are more efficient than floating point (see the table below). Also, there already exists dedicated hardware for deep learning that uses integer operations (such as the 1st generation of Google TPU) so understanding the best way to use integers is well-motivated.<br />
{| class="wikitable"<br />
|+Rough Energy Costs in 45nm 0.9V [1]<br />
!<br />
! colspan="2" |Energy(pJ)<br />
! colspan="2" |Area(<math>\mu m^2</math>)<br />
|-<br />
!Operation<br />
!MUL<br />
!ADD<br />
!MUL<br />
!ADD<br />
|-<br />
|8-bit INT<br />
|0.2<br />
|0.03<br />
|282<br />
|36<br />
|-<br />
|16-bit FP<br />
|1.1<br />
|0.4<br />
|1640<br />
|1360<br />
|-<br />
|32-bit FP<br />
|3.7<br />
|0.9<br />
|7700<br />
|4184<br />
|}<br />
The authors call the framework WAGE because they consider how best to handle the '''W'''eights, '''A'''ctivations, '''G'''radients, and '''E'''rrors separately.<br />
<br />
== Related Work ==<br />
<br />
=== Weight and Activation ===<br />
Existing works to train DNNs on binary weights and activations [2] add noise to weights and activations as a form of regularization. The use of high-precision accumulation is required for SGD optimization since real-valued gradients are obtained from real-valued variables. XNOR-Net [11] uses bitwise operations to approximate convolutions in a highly memory-efficient manner, and applies a filter-wise scaling factor for weights to improve performance. However, these floating-point factors are calculated simultaneously during training, which aggravates the training effort. Ternary weight networks (TWN) [3] and Trained ternary quantization (TTQ) [9] offer more expressive ability than binary weight networks by constraining the weights to be ternary-valued {-1,0,1} using two symmetric thresholds.<br />
<br />
=== Gradient Computation and Accumulation ===<br />
The DoReFa-Net quantizes gradients to low-bandwidth floating point numbers with discrete states in the backwards pass. In order to reduce the overhead of gradient synchronization in distributed training the TernGrad method quantizes the gradient updates to ternary values. In both works the weights are still stored and updated with float32, and the quantization of batch normalization and its derivative is ignored.<br />
<br />
== WAGE Quantization ==<br />
The core idea of the proposed method is to constrain the following to low-bitwidth integers on each layer:<br />
* '''W:''' weight in inference<br />
* '''a:''' activation in inference<br />
* '''e:''' error in backpropagation<br />
* '''g:''' gradient in backpropagation<br />
[[File:p32fig1.PNG|center|thumb|800px|Four operators QW (·), QA(·), QG(·), QE(·) added in WAGE computation dataflow to reduce precision, bitwidth of signed integers are below or on the right of arrows, activations are included in MAC for concision.]]<br />
The error and gradient are defined as:<br />
<br />
<math>e^i = \frac{\partial L}{\partial a^i}, g^i = \frac{\partial L}{\partial W^i}</math><br />
<br />
where L is the loss function.<br />
<br />
The precision in bits of the errors, activations, gradients, and weights are <math>k_E</math>, <math>k_A</math>, <math>k_G</math>, and <math>k_W</math> respectively. As shown in the above figure, each quantity also has a quantization operators to reduce bitwidth increases caused by multiply-accumulate (MAC) operations. Also, note that since this is a layer-by-layer approach, each layer may be followed or preceded by a layer with different precision, or even a layer using floating point math.<br />
<br />
=== Shift-Based Linear Mapping and Stochastic Mapping ===<br />
The proposed method makes use of a linear mapping where continuous, unbounded values are discretized for each bitwidth <math>k</math> with a uniform spacing of<br />
<br />
<math>\sigma(k) = 2^{1-k}, k \in Z_+ </math><br />
With this, the full quantization function is<br />
<br />
<math>Q(x,k) = Clip\left \{ \sigma(k) \cdot round\left [ \frac{x}{\sigma(k)} \right ], -1 + \sigma(k), 1 - \sigma(k) \right \}</math>, <br />
<br />
where <math>round</math> approximates continuous values to their nearest discrete state, and <math>Clip</math> is the saturation function that clips unbounded values to <math>[-1 + \sigma, 1 - \sigma]</math>. Note that this function is only using when simulating integer operations on floating-point hardware, on native integer hardware, this is done automatically. In addition to this quantization function, a distribution scaling factor is used in some quantization operators to preserve as much variance as possible when applying the quantization function above. The scaling factor is defined below.<br />
<br />
<math>Shift(x) = 2^{round(log_2(x))}</math><br />
<br />
Finally, stochastic rounding is substituted for small or real-valued updates during gradient accumulation.<br />
<br />
A visual representation of these operations is below.<br />
[[File:p32fig2.PNG|center|thumb|800px|Quantization methods used in WAGE. The notation <math>P, x, \lfloor \cdot \rfloor, \lceil \cdot \rceil</math> denotes probability, vector, floor and ceil, respectively. <math>Shift(\cdot)</math> refers to distribution shifting with a certain argument]]<br />
<br />
=== Weight Initialization ===<br />
In this work, batch normalization is simplified to a constant scaling layer in order to sidestep the problem of normalizing outputs without floating point math, and to remove the extra memory requirement with batch normalization. As such, some care must be taken when initializing weights. The authors use a modified initialization method base on MSRA [4].<br />
<br />
<math>W \thicksim U(-L, +L),L = max \left \{ \sqrt{6/n_{in}}, L_{min} \right \}, L_{min} = \beta \sigma</math><br />
<br />
<math>n_{in}</math> is the layer fan-in number, <math>U</math> denotes uniform distribution. The original initialization method for <math>\eta</math> is modified by adding the condition that the distribution width should be at least <math>\beta \sigma</math>, where <math>\beta</math> is a constant greater than 1 and <math>\sigma</math> is the minimum step size seen already. This prevents weights being initialised to all-zeros in the case where the bitwidth is low, or the fan-in number is high.<br />
<br />
=== Quantization Details ===<br />
<br />
==== Weight <math>Q_W(\cdot)</math> ====<br />
<math>W_q = Q_W(W) = Q(W, k_W)</math><br />
<br />
The quantization operator is simply the quantization function previously introduced. <br />
<br />
==== Activation <math>Q_A(\cdot)</math> ====<br />
The authors say that the variance of the weights passed through this function will be scaled compared to the variance of the weights as initialized. To prevent this effect from blowing up the network outputs, they introduce a scaling factor <math>\alpha</math>. Notice that it is constant for each layer.<br />
<br />
<math>\alpha = max \left \{ Shift(L_{min} / L), 1 \right \}</math><br />
<br />
The quantization operator is then<br />
<br />
<math>a_q = Q_A(a) = Q(a/\alpha, k_A)</math><br />
<br />
The scaling factor approximates batch normalization.<br />
<br />
==== Error <math>Q_E(\cdot)</math> ====<br />
The magnitude of the error can vary greatly, and that a previous approach (DoReFa-Net [5]) solves the issue by using an affine transform to map the error to the range <math>[-1, 1]</math>, apply quantization, and then applying the inverse transform. However, the authors claim that this approach still requires using float32, and that the magnitude of the error is unimportant: rather it is the orientation of the error. Thus, they only scale the error distribution to the range <math>\left [ -\sqrt2, \sqrt2 \right ]</math> and quantise:<br />
<br />
<math>e_q = Q_E(e) = Q(e/Shift(max\{|e|\}), k_E)</math><br />
<br />
Max is the element-wise maximum. Note that this discards any error elements less than the minimum step size.<br />
<br />
==== Gradient <math>Q_G(\cdot)</math> ====<br />
Similar to the activations and errors, the gradients are rescaled:<br />
<br />
<math>g_s = \eta \cdot g/Shift(max\{|g|\})</math><br />
<br />
<math> \eta </math> is a shift-based learning rate. It is an integer power of 2. The shifted gradients are represented in units of minimum step sizes <math> \sigma(k) </math>. When reducing the bitwidth of the gradients (remember that the gradients are coming out of a MAC operation, so the bitwidth may have increased) stochastic rounding is used as a substitute for small gradient accumulation.<br />
<br />
<math>\Delta W = Q_G(g) = \sigma(k_G) \cdot sgn(g_s) \cdot \left \{ \lfloor | g_s | \rfloor + Bernoulli(|g_s|<br />
- \lfloor | g_s | \rfloor) \right \}</math><br />
<br />
This randomly rounds the result of the MAC operation up or down to the nearest quantization for the given gradient bitwidth. The weights are updated with the resulting discrete increments:<br />
<br />
<math>W_{t+1} = Clip \left \{ W_t - \Delta W_t, -1 + \sigma(k_G), 1 - \sigma(k_G) \right \}</math><br />
<br />
=== Miscellaneous ===<br />
To train WAGE networks, the authors used pure SGD exclusively because more complicated techniques such as Momentum or RMSProp increase memory consumption and are complicated by the rescaling that happens within each quantization operator.<br />
<br />
The quantization and stochastic rounding are a form of regularization.<br />
<br />
The authors didn't use a traditional softmax with cross-entropy loss for the experiments because there does not yet exist a softmax layer for low-bit integers. Instead, they use a sum of squared error loss. This works for tasks with a small number of categories, but does not scale well.<br />
<br />
== Experiments ==<br />
For all experiments, the default layer bitwidth configuration is 2-8-8-8 for Weights, Activations, Gradients, and Error bits. The weight bitwidth is set to 2 because that results in ternary weights, and therefore no multiplication during inference. They authors argue that the bitwidth for activation and errors should be the same because the computation graph for each is similar and might use the same hardware. During training, the weight bitwidth is 8. For inference the weights are ternarized.<br />
<br />
=== Implementation Details ===<br />
MNIST: Network is LeNet-5 variant [6] with 32C5-MP2-64C5-MP2-512FC-10SSE.<br />
<br />
SVHN & CIFAR10: VGG variant [7] with 2×(128C3)-MP2-2×(256C3)-MP2-2×(512C3)-MP2-1024FC-10SSE. For CIFAR10 dataset, the data augmentation is followed in Lee et al. (2015) [10] for training.<br />
<br />
ImageNet: AlexNet variant [8] on ILSVRC12 dataset.<br />
{| class="wikitable"<br />
|+Test or validation error rates (%) in previous works and WAGE on multiple datasets. Opt denotes gradient descent optimizer, withM means SGD with momentum, BN represents batch normalization, 32 bit refers to float32, and ImageNet top-k format: top1/top5.<br />
!Method<br />
!<math>k_W</math><br />
!<math>k_A</math><br />
!<math>k_G</math><br />
!<math>k_E</math><br />
!Opt<br />
!BN<br />
!MNIST<br />
!SVHN<br />
!CIFAR10<br />
!ImageNet<br />
|-<br />
|BC<br />
|1<br />
|32<br />
|32<br />
|32<br />
|Adam<br />
|yes<br />
|1.29<br />
|2.30<br />
|9.90<br />
|<br />
|-<br />
|BNN<br />
|1<br />
|1<br />
|32<br />
|32<br />
|Adam<br />
|yes <br />
|0.96<br />
|2.53<br />
|10.15<br />
|<br />
|-<br />
|BWN<br />
|1<br />
|32<br />
|32<br />
|32<br />
|withM<br />
|yes<br />
|<br />
|<br />
|<br />
|43.2/20.6<br />
|-<br />
|XNOR<br />
|1<br />
|1<br />
|32<br />
|32<br />
|Adam<br />
|yes<br />
|<br />
|<br />
|<br />
|55.8/30.8<br />
|-<br />
|TWN<br />
|2<br />
|32<br />
|32<br />
|32<br />
|withM<br />
|yes<br />
|0.65<br />
|<br />
|7.44<br />
|'''34.7/13.8'''<br />
|-<br />
|TTQ<br />
|2<br />
|32<br />
|32<br />
|32<br />
|Adam<br />
|yes<br />
|<br />
|<br />
|6.44<br />
|42.5/20.3<br />
|-<br />
|DoReFa<br />
|8<br />
|8<br />
|32<br />
|8<br />
|Adam<br />
|yes<br />
|<br />
|2.30<br />
|<br />
|47.0/<br />
|-<br />
|TernGrad<br />
|32<br />
|32<br />
|2<br />
|32<br />
|Adam<br />
|yes<br />
|<br />
|<br />
|14.36<br />
|42.4/19.5<br />
|-<br />
|WAGE<br />
|2<br />
|8<br />
|8<br />
|8<br />
|SGD<br />
|no<br />
|'''0.40'''<br />
|'''1.92'''<br />
|'''6.78'''<br />
|51.6/27.8<br />
|}<br />
<br />
=== Training Curves and Regularization ===<br />
The authors compare the 2-8-8-8 WAGE configuration introduced above, a 2-8-f-f (meaning float32) configuration, and a completely floating point version on CIFAR10. The test error is plotted against epoch. For training these networks, the learning rate is divided by 8 at the 200th epoch and again at the 250th epoch.<br />
[[File:p32fig3.PNG|center|thumb|800px|Training curves of WAGE variations and a vanilla CNN on CIFAR10]]<br />
The convergence of the 2-8-8-8 has comparable convergence to the vanilla CNN and outperforms the 2-8-f-f variant. The authors speculate that this is because the extra discretization acts as a regularizer.<br />
<br />
=== Bitwidth of Errors ===<br />
The CIFAR10 test accuracy is plotted against bitwidth below and the error density for a single layer is compared with the Vanilla network.<br />
[[File:p32fig4.PNG|center|thumb|520x522px|The 10 run accuracies of different <math>k_E</math>]]<br />
<br />
[[File:32_error.png|center|thumb|520x522px|Histogram of errors for Vanilla network and Wage network. After being quantized and shifted each layer, the error is reshaped and so most orientation information is retained. ]]<br />
<br />
The table below shows the test error rates on CIFAR10 when left-shift upper boundary with factor γ. From this table we could see that large values play critical roles for backpropagation training even though they don't have many while the majority with small values actually just noises.<br />
<br />
[[File:testerror_rate.png|center]]<br />
<br />
=== Bitwidth of Gradients ===<br />
<br />
The authors next investigated the choice of a proper <math>k_G</math> for gradients using the CIFAR10 dataset. <br />
<br />
{| class="wikitable"<br />
|+Test error rates (%) on CIFAR10 with different <math>k_G</math><br />
!<math>k_G</math><br />
!2<br />
!3<br />
!4<br />
!5<br />
!6<br />
!7<br />
!8<br />
!9<br />
!10<br />
!11<br />
!12<br />
|-<br />
|error<br />
|54.22<br />
|51.57<br />
|28.22<br />
|18.01<br />
|11.48<br />
|7.61<br />
|6.78<br />
|6.63<br />
|6.43<br />
|6.55<br />
|6.57<br />
|}<br />
<br />
The results show similar bitwidth requirements as the last experiment for <math>k_E</math>.<br />
<br />
The authors also examined the effect of bitwidth on the ImageNet implementation.<br />
<br />
Here, C denotes 12 bits (Hexidecimal) and BN refers to batch normalization being added. 7 models are used: 2888 from the first experiment, 288C for more accurate errors (12 bits), 28C8 for larger buffer space, 28f8 for non-quantization of gradients, 28ff for errors and gradients in float32, and 28ff with BN added. The baseline vanilla model refers to the original AlexNet architecture. <br />
<br />
{| class="wikitable"<br />
|+Top-5 error rates (%) on ImageNet with different <math>k_G</math>and <math>k_E</math><br />
!Pattern<br />
!vanilla<br />
!28ff-BN<br />
!28ff<br />
!28f8<br />
!28C8<br />
!288C<br />
!2888<br />
|-<br />
|error<br />
|19.29<br />
|20.67<br />
|24.14<br />
|23.92<br />
|26.88<br />
|28.06<br />
|27.82<br />
|}<br />
<br />
The comparison between 28C8 and 288C shows that the model may perform better if it has more buffer space <math>k_G</math> for gradient accumulation than if it has high-resolution orientation <math>k_E</math>. The authors also noted that batch normalization and <math>k_G</math> are more important for ImageNet because the training set samples are highly variant.<br />
<br />
== Discussion ==<br />
The authors have a few areas they believe this approach could be improved.<br />
<br />
'''MAC Operation:''' The 2-8-8-8 configuration was chosen because the low weight bitwidth means there aren't any multiplication during inference. However, this does not remove the requirement for multiplication during training. 2-2-8-8 configuration satisfies this requirement, but it is difficult to train and detrimental to the accuracy.<br />
<br />
'''Non-linear Quantization:''' The linear mapping used in this approach is simple, but there might be a more effective mapping. For example, a logarithmic mapping could be more effective if the weights and activations have a log-normal distribution.<br />
<br />
'''Normalization:''' Normalization layers (softmax, batch normalization) were not used in this paper. Quantized versions are an area of future work<br />
<br />
== Conclusion ==<br />
<br />
A framework for training and inference without the use of floating-point representation is presented. By quantizing all operations and operands of a network, the authors successfully reduce the energy costs of both training and inference with deep learning architectures. Future work may further improve compression and memory requirements.<br />
<br />
== References ==<br />
<br />
# Sze, Vivienne; Chen, Yu-Hsin; Yang, Tien-Ju; Emer, Joel (2017-03-27). [http://arxiv.org/abs/1703.09039 "Efficient Processing of Deep Neural Networks: A Tutorial and Survey"]. arXiv:1703.09039 [cs].<br />
# Courbariaux, Matthieu; Bengio, Yoshua; David, Jean-Pierre (2015-11-01). [http://arxiv.org/abs/1511.00363 "BinaryConnect: Training Deep Neural Networks with binary weights during propagations"]. arXiv:1511.00363 [cs].<br />
# Li, Fengfu; Zhang, Bo; Liu, Bin (2016-05-16). [http://arxiv.org/abs/1605.04711 "Ternary Weight Networks"]. arXiv:1605.04711 [cs].<br />
# He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2015-02-06). [http://arxiv.org/abs/1502.01852 "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification"]. arXiv:1502.01852 [cs].<br />
# Zhou, Shuchang; Wu, Yuxin; Ni, Zekun; Zhou, Xinyu; Wen, He; Zou, Yuheng (2016-06-20). [http://arxiv.org/abs/1606.06160 "DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients"]. arXiv:1606.06160 [cs].<br />
# Lecun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. (November 1998). [http://ieeexplore.ieee.org/document/726791/?reload=true "Gradient-based learning applied to document recognition"]. Proceedings of the IEEE. 86 (11): 2278–2324. doi:10.1109/5.726791. ISSN 0018-9219.<br />
# Simonyan, Karen; Zisserman, Andrew (2014-09-04). [http://arxiv.org/abs/1409.1556 "Very Deep Convolutional Networks for Large-Scale Image Recognition"]. arXiv:1409.1556 [cs].<br />
# Krizhevsky, Alex; Sutskever, Ilya; Hinton, Geoffrey E (2012). Pereira, F.; Burges, C. J. C.; Bottou, L.; Weinberger, K. Q., eds. [http://papers.nips.cc/paper/4824-imagenet-classification-with-deep-convolutional-neural-networks.pdf Advances in Neural Information Processing Systems 25 (PDF)]. Curran Associates, Inc. pp. 1097–1105.<br />
# Chenzhuo Zhu, Song Han, Huizi Mao, and William J Dally. Trained ternary quantization. arXiv preprint arXiv:1612.01064, 2016.<br />
# Chen-Yu Lee, Saining Xie, Patrick Gallagher, Zhengyou Zhang, and Zhuowen Tu. Deeplysupervisednets. In Artificial Intelligence and Statistics, pp. 562–570, 2015.<br />
# Mohammad Rastegari, Vicente Ordonez, Joseph Redmon, and Ali Farhadi. Xnor-net: Imagenet classification using binary convolutional neural networks. In European Conference on Computer Vision, pp. 525–542. Springer, 2016.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Dynamic_Routing_Between_Capsules_STAT946&diff=36057Dynamic Routing Between Capsules STAT9462018-04-03T05:42:55Z<p>J3ku: Update "capsule" definition</p>
<hr />
<div>= Presented by =<br />
<br />
Yang, Tong(Richard)<br />
<br />
= Contributions =<br />
<br />
This paper introduces the concept of "capsules" and an approach to implement its concept in neural networks. Capsules are a group of neurons used to represent various properties of an entity/object present in the image, such as pose, deformation, and even the existence of the entity. Instead of the obvious representation of a logistic unit for the probability of existence, the paper explores using the length of the capsule output vector to represent existence, and the orientation to represent other properties of the entity. The paper has the following major contributions:<br />
<br />
* Proposed an alternative approach to max-pooling, which is called routing-by-agreement.<br />
* Demonstrated an mathematical structure for capsule layers and routing mechanism that builds a prototype architecture for capsule networks. <br />
* Presented the promising results of CapsNet that confirms its value as a new direction for development in deep learning.<br />
<br />
= Hinton's Critiques on CNN =<br />
<br />
In the past talk, Hinton tried to explained why max-pooling is the biggest problem in current convolutional network structure, here are some highlights from his talk. <br />
<br />
== Four arguments against pooling ==<br />
<br />
* It is a bad fit to the psychology of shape perception: It does not explain why we assign intrinsic coordinate frames to objects and why they have such huge effects.<br />
<br />
* It solves the wrong problem: We want equivariance, not invariance. Disentangling rather than discarding.<br />
<br />
* It fails to use the underlying linear structure: It does not make use of the natural linear manifold that perfectly handles the largest source of variance in images.<br />
<br />
* Pooling is a poor way to do dynamic routing: We need to route each part of the input to the neurons that know how to deal with it. Finding the best routing is equivalent to parsing the image.<br />
<br />
===Intuition Behind Capsules ===<br />
We try to achieve viewpoint invariance in the activities of neurons by doing max-pooling. Invariance here means that by changing the input a little, the output still stays the same while the activity is just the output signal of a neuron. In other words, when in the input image we shift the object that we want to detect by a little bit, networks activities (outputs of neurons) will not change because of max pooling and the network will still detect the object. But the spacial relationships are not taken care of in this approach so instead capsules are used, because they encapsulate all important information about the state of the features they are detecting in a form of a vector. Capsules encode probability of detection of a feature as the length of their output vector. And the state of the detected feature is encoded as the direction in which that vector points to. So when detected feature moves around the image or its state somehow changes, the probability still stays the same (length of vector does not change), but its orientation changes.<br />
<br />
== Equivariance ==<br />
<br />
To deal with the invariance problem of CNN, Hinton proposes the concept called equivariance, which is the foundation of capsule concept.<br />
<br />
=== Two types of equivariance ===<br />
<br />
==== Place-coded equivariance ====<br />
If a low-level part moves to a very different position it will be represented by a different capsule.<br />
<br />
==== Rate-coded equivariance ====<br />
If a part only moves a small distance it will be represented by the same capsule but the pose outputs of the capsule will change.<br />
<br />
Higher-level capsules have bigger domains so low-level place-coded equivariance gets converted into high-level rate-coded equivariance.<br />
<br />
= Dynamic Routing =<br />
<br />
In the second section of this paper, authors give a mathematical representations for two key features in routing algorithm in capsule network, which are squashing and agreement. The general setting for this algorithm is between two arbitrary capsules i and j. Capsule j is assumed to be an arbitrary capsule from the first layer of capsules, and capsule i is an arbitrary capsule from the layer below. The purpose of routing algorithm is generate a vector output for routing decision between capsule j and capsule i. Furthermore, this vector output will be used in the decision for choice of dynamic routing. <br />
<br />
== Routing Algorithm ==<br />
<br />
The routing algorithm is as the following:<br />
<br />
[[File:DRBC_Figure_1.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
In the following sections, each part of this algorithm will be explained in details.<br />
<br />
=== Log Prior Probability ===<br />
<br />
<math>b_{ij}</math> represents the log prior probabilities that capsule i should be coupled to capsule j, and updated in each routing iteration. As line 2 suggests, the initial values of <math>b_{ij}</math> for all possible pairs of capsules are set to 0. In the very first routing iteration, <math>b_{ij}</math> equals to zero. For each routing iteration, <math>b_{ij}</math> gets updated by the value of agreement, which will be explained later.<br />
<br />
=== Coupling Coefficient === <br />
<br />
<math>c_{ij}</math> represents the coupling coefficient between capsule j and capsule i. It is calculated by applying the softmax function on the log prior probability <math>b_{ij}</math>. The mathematical transformation is shown below (Equation 3 in paper): <br />
<br />
\begin{align}<br />
c_{ij} = \frac{exp(b_ij)}{\sum_{k}exp(b_ik)}<br />
\end{align}<br />
<br />
<math>c_{ij}</math> are served as weights for computing the weighted sum and probabilities. Therefore, as probabilities, they have the following properties:<br />
<br />
\begin{align}<br />
c_{ij} \geq 0, \forall i, j<br />
\end{align}<br />
<br />
and, <br />
<br />
\begin{align}<br />
\sum_{i,j}c_{ij} = 1, \forall i, j<br />
\end{align}<br />
<br />
=== Predicted Output from Layer Below === <br />
<br />
<math>u_{i}</math> are the output vector from capsule i in the lower layer, and <math>\hat{u}_{j|i}</math> are the input vector for capsule j, which are the "prediction vectors" from the capsules in the layer below. <math>\hat{u}_{j|i}</math> is produced by multiplying <math>u_{i}</math> by a weight matrix <math>W_{ij}</math>, such as the following:<br />
<br />
\begin{align}<br />
\hat{u}_{j|i} = W_{ij}u_i<br />
\end{align}<br />
<br />
where <math>W_{ij}</math> encodes some spatial relationship between capsule j and capsule i.<br />
<br />
=== Capsule ===<br />
<br />
By using the definitions from previous sections, the total input vector for an arbitrary capsule j can be defined as:<br />
<br />
\begin{align}<br />
s_j = \sum_{i}c_{ij}\hat{u}_{j|i}<br />
\end{align}<br />
<br />
which is a weighted sum over all prediction vectors by using coupling coefficients.<br />
<br />
=== Squashing ===<br />
<br />
The length of <math>s_j</math> is arbitrary, which is needed to be addressed with. The next step is to convert its length between 0 and 1, since we want the length of the output vector of a capsule to represent the probability that the entity represented by the capsule is present in the current input. The "squashing" process is shown below:<br />
<br />
\begin{align}<br />
v_j = \frac{||s_j||^2}{1+||s_j||^2}\frac{s_j}{||s_j||}<br />
\end{align}<br />
<br />
Notice that "squashing" is not just normalizing the vector into unit length. In addition, it does extra non-linear transformation to ensure that short vectors get shrunk to almost zero length and long vectors get shrunk to a length slightly below 1. The reason for doing this is to make decision of routing, which is called "routing by agreement" much easier to make between capsule layers.<br />
<br />
=== Agreement ===<br />
<br />
The final step of a routing iteration is to form an routing agreement <math>a_{ij}</math>, which is represents as a scalar product:<br />
<br />
\begin{align}<br />
a_{ij} = v_{j}\hat{u}_{j|i}<br />
\end{align}<br />
<br />
As we mentioned in "squashing" section, the length of <math>v_{j}</math> is either close to 0 or close to 1, which will effect the magnitude of <math>a_{ij}</math> in this case. Therefore, the magnitude of <math>a_{ij}</math> indicate the how strong the routing algorithm agrees on taking the route between capsule j and capsule i. For each routing iteration, the log prior probability, <math>b_{ij}</math> will be updated by adding the value of its agreement value, which will effect how the coupling coefficients are computed in the next routing iteration. Because of the "squashing" process, we will eventually end up with a capsule j with its <math>v_{j}</math> close to 1 while all other capsules with its <math>v_{j}</math> close to 0, which indicates that this capsule j should be activated.<br />
<br />
= CapsNet Architecture =<br />
<br />
The second part of this paper discuss the experiment results from a 3-layer CapsNet, the architecture can be divided into two parts, encoder and decoder. <br />
<br />
== Encoder == <br />
<br />
[[File:DRBC_Architecture.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
=== How many routing iteration to use? === <br />
In appendix A of this paper, the authors have shown the empirical results from 500 epochs of training at different choice of routing iterations. According to their observation, more routing iterations increases the capacity of CapsNet but tends to bring additional risk of overfitting. Moreover, CapsNet with routing iterations less than three are not effective in general. As result, they suggest 3 iterations of routing for all experiments.<br />
<br />
=== Marginal loss for digit existence ===<br />
<br />
The experiments performed include segmenting overlapping digits on MultiMINST data set, so the loss function has be adjusted for presents of multiple digits. The marginal lose <math>L_k</math> for each capsule k is calculate by:<br />
<br />
\begin{align}<br />
L_k = T_k max(0, m^+ - ||v_k||)^2 + \lambda(1 - T_k) max(0, ||v_k|| - m^-)^2<br />
\end{align}<br />
<br />
where <math>m^+ = 0.9</math>, <math>m^- = 0.1</math>, and <math>\lambda = 0.5</math>.<br />
<br />
<math>T_k</math> is an indicator for presence of digit of class k, it takes value of 1 if and only if class k is presented. If class k is not presented, <math>\lambda</math> down-weight the loss which shrinks the lengths of the activity vectors for all the digit capsules. By doing this, The loss function penalizes the initial learning for all absent digit class, since we would like the top-level capsule for digit class k to have long instantiation vector if and only if that digit class is present in the input.<br />
<br />
=== Layer 1: Conv1 === <br />
<br />
The first layer of CapsNet. Similar to CNN, this is just convolutional layer that converts pixel intensities to activities of local feature detectors. <br />
<br />
* Layer Type: Convolutional Layer.<br />
* Input: <math>28 \times 28</math> pixels.<br />
* Kernel size: <math>9 \times 9</math>.<br />
* Number of Kernels: 256.<br />
* Activation function: ReLU.<br />
* Output: <math>20 \times 20 \times 256</math> tensor.<br />
<br />
=== Layer 2: PrimaryCapsules ===<br />
<br />
The second layer is formed by 32 primary 8D capsules. By 8D, it means that each primary capsule contains 8 convolutional units with a <math>9 \times 9</math> kernel and a stride of 2. Each capsule will take a <math>20 \times 20 \times 256</math> tensor from Conv1 and produce an output of a <math>6 \times 6 \times 8</math> tensor.<br />
<br />
* Layer Type: Convolutional Layer<br />
* Input: <math>20 \times 20 \times 256</math> tensor.<br />
* Number of capsules: 32.<br />
* Number of convolutional units in each capsule: 8.<br />
* Size of each convolutional unit: <math>6 \times 6</math>.<br />
* Output: <math>6 \times 6 \times 8</math> 8-dimensional vectors.<br />
<br />
=== Layer 3: DigitsCaps ===<br />
<br />
The last layer has 10 16D capsules, one for each digit. Not like the PrimaryCapsules layer, this layer is fully connected. Since this is the top capsule layer, dynamic routing mechanism will be applied between DigitsCaps and PrimaryCapsules. The process begins by taking a transformation of predicted output from PrimaryCapsules layer. Each output is a 8-dimensional vector, which needed to be mapped to a 16-dimensional space. Therefore, the weight matrix, <math>W_{ij}</math> is a <math>8 \times 16</math> matrix. The next step is to acquire coupling coefficients from routing algorithm and to perform "squashing" to get the output. <br />
<br />
* Layer Type: Fully connected layer.<br />
* Input: <math>6 \times 6 \times 8</math> 8-dimensional vectors.<br />
* Output: <math>16 \times 10 </math> matrix.<br />
<br />
=== The loss function ===<br />
<br />
The output of the loss function would be a ten-dimensional one-hot encoded vector with 9 zeros and 1 one at the correct position.<br />
<br />
<br />
== Regularization Method: Reconstruction ==<br />
<br />
This is regularization method introduced in the implementation of CapsNet. The method is to introduce a reconstruction loss (scaled down by 0.0005) to margin loss during training. The authors argue this would encourage the digit capsules to encode the instantiation parameters the input digits. All the reconstruction during training is by using the true labels of the image input. The results from experiments also confirms that adding the reconstruction regularizer enforces the pose encoding in CapsNet and thus boots the performance of routing procedure. <br />
<br />
=== Decoder ===<br />
<br />
The decoder consists of 3 fully connected layers, each layer maps pixel intensities to pixel intensities. The number of parameters in each layer and the activation functions used are indicated in the figure below:<br />
<br />
[[File:DRBC_Decoder.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
=== Result ===<br />
<br />
The authors includes some results for CapsNet classification test accuracy to justify the result of reconstruction. We can see that for CapsNet with 1 routing iteration and CapsNet with 3 routing iterations, implement reconstruction shows significant improvements in both MINIST and MultiMINST data set. These improvements show the importance of routing and reconstruction regularizer. <br />
<br />
[[File:DRBC_Reconstruction.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
= Experiment Results for CapsNet = <br />
<br />
In this part, the authors demonstrate experiment results of CapsNet on different data sets, such as MINIST and different variation of MINST, such as expanded MINST, affNIST, MultiMNIST. Moreover, they also briefly discuss the performance on some other popular data set such CIFAR 10. <br />
<br />
== MINST ==<br />
<br />
=== Highlights ===<br />
<br />
* CapsNet archives state-of-the-art performance on MINST.<br />
* CapsNet with shallow structure (3 layers) achieves performance that only achieves by deeper network before.<br />
<br />
=== Interpretation of Each Capsule ===<br />
<br />
The authors suggest that they found evidence that dimension of some capsule always captures some variance of the digit, while some others represents the global combinations of different variations, this would open some possibility for interpretation of capsules in the future. Some results from perturbations are shown below: <br />
<br />
[[File:DRBC_Dimension.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
== affNIST == <br />
<br />
affNIT data set contains different affine transformation of original MINST data set. By the concept of capsule, CapsNet should gain more robustness from its equivariance nature, and the result confirms this. Compare the baseline CNN, CapsNet achieves 13% improvement on accuracy.<br />
<br />
== MultiMNIST ==<br />
<br />
The MultiMNIST is basically the overlapped version of MINIST. An important point to notice here is that this data set is generated by overlaying a digit on top of another digit from the same set but different class. In other words, the case of stacking digits from the same class is not allowed in MultiMINST. For example, stacking a 5 on a 0 is allowed, but stacking a 5 on another 5 is not. The reason is that CapsNet suffers from the "crowding" effect which will be discussed in the weakness of CapsNet section. <br />
<br />
== Other data sets ==<br />
<br />
CapsNet is used on other data sets such as CIFAR1-, smallNORB and SVHN. The results are not comparable with state-of-the-art performance, but it is still promising since this architecture is the very first, while other networks have been development for a long time.<br />
<br />
= Conclusion = <br />
<br />
This paper discuss the specific part of capsule network, which is the routing-by-agreement mechanism. The authors suggest this is a great approach to solve the current problem with max-pooling in convolutional neural network. Moreover, as author mentioned, the approach mentioned in this paper is only one possible implementation of the capsule concept. The preliminary results from experiment using a simple shallow CapsNet also demonstrate unparalleled performance that indicates the capsules are a direction worth exploring. <br />
<br />
= Weakness of Capsule Network =<br />
<br />
* Routing algorithm introduces internal loops for each capsule. As number of capsules and layers increases, these internal loops may exponentially expand the training time. <br />
* Capsule network suffers a perceptual phenomenon called "crowding", which is common for human vision as well. To address this weakness, capsules have to make a very strong representation assumption that at each location of the image, there is at most one instance of the type of entity that capsule represents. This is also the reason for not allowing overlaying digits from same class in generating process of MultiMINST.<br />
* Other criticisms include that the design of capsule networks requires domain knowledge or feature engineering, contrary to the abstraction-oriented goals of deep learning.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Dynamic_Routing_Between_Capsules_STAT946&diff=36056Dynamic Routing Between Capsules STAT9462018-04-03T05:38:33Z<p>J3ku: Add "capsule" definition</p>
<hr />
<div>= Presented by =<br />
<br />
Yang, Tong(Richard)<br />
<br />
= Contributions =<br />
<br />
This paper introduces the concept of "capsules" and an approach to implement its concept in neural networks. Capsules are a group of neurons used to represent various properties of an entity/object present in the image. The paper has the following major contributions:<br />
<br />
* Proposed an alternative approach to max-pooling, which is called routing-by-agreement.<br />
* Demonstrated an mathematical structure for capsule layers and routing mechanism that builds a prototype architecture for capsule networks. <br />
* Presented the promising results of CapsNet that confirms its value as a new direction for development in deep learning.<br />
<br />
= Hinton's Critiques on CNN =<br />
<br />
In the past talk, Hinton tried to explained why max-pooling is the biggest problem in current convolutional network structure, here are some highlights from his talk. <br />
<br />
== Four arguments against pooling ==<br />
<br />
* It is a bad fit to the psychology of shape perception: It does not explain why we assign intrinsic coordinate frames to objects and why they have such huge effects.<br />
<br />
* It solves the wrong problem: We want equivariance, not invariance. Disentangling rather than discarding.<br />
<br />
* It fails to use the underlying linear structure: It does not make use of the natural linear manifold that perfectly handles the largest source of variance in images.<br />
<br />
* Pooling is a poor way to do dynamic routing: We need to route each part of the input to the neurons that know how to deal with it. Finding the best routing is equivalent to parsing the image.<br />
<br />
===Intuition Behind Capsules ===<br />
We try to achieve viewpoint invariance in the activities of neurons by doing max-pooling. Invariance here means that by changing the input a little, the output still stays the same while the activity is just the output signal of a neuron. In other words, when in the input image we shift the object that we want to detect by a little bit, networks activities (outputs of neurons) will not change because of max pooling and the network will still detect the object. But the spacial relationships are not taken care of in this approach so instead capsules are used, because they encapsulate all important information about the state of the features they are detecting in a form of a vector. Capsules encode probability of detection of a feature as the length of their output vector. And the state of the detected feature is encoded as the direction in which that vector points to. So when detected feature moves around the image or its state somehow changes, the probability still stays the same (length of vector does not change), but its orientation changes.<br />
<br />
== Equivariance ==<br />
<br />
To deal with the invariance problem of CNN, Hinton proposes the concept called equivariance, which is the foundation of capsule concept.<br />
<br />
=== Two types of equivariance ===<br />
<br />
==== Place-coded equivariance ====<br />
If a low-level part moves to a very different position it will be represented by a different capsule.<br />
<br />
==== Rate-coded equivariance ====<br />
If a part only moves a small distance it will be represented by the same capsule but the pose outputs of the capsule will change.<br />
<br />
Higher-level capsules have bigger domains so low-level place-coded equivariance gets converted into high-level rate-coded equivariance.<br />
<br />
= Dynamic Routing =<br />
<br />
In the second section of this paper, authors give a mathematical representations for two key features in routing algorithm in capsule network, which are squashing and agreement. The general setting for this algorithm is between two arbitrary capsules i and j. Capsule j is assumed to be an arbitrary capsule from the first layer of capsules, and capsule i is an arbitrary capsule from the layer below. The purpose of routing algorithm is generate a vector output for routing decision between capsule j and capsule i. Furthermore, this vector output will be used in the decision for choice of dynamic routing. <br />
<br />
== Routing Algorithm ==<br />
<br />
The routing algorithm is as the following:<br />
<br />
[[File:DRBC_Figure_1.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
In the following sections, each part of this algorithm will be explained in details.<br />
<br />
=== Log Prior Probability ===<br />
<br />
<math>b_{ij}</math> represents the log prior probabilities that capsule i should be coupled to capsule j, and updated in each routing iteration. As line 2 suggests, the initial values of <math>b_{ij}</math> for all possible pairs of capsules are set to 0. In the very first routing iteration, <math>b_{ij}</math> equals to zero. For each routing iteration, <math>b_{ij}</math> gets updated by the value of agreement, which will be explained later.<br />
<br />
=== Coupling Coefficient === <br />
<br />
<math>c_{ij}</math> represents the coupling coefficient between capsule j and capsule i. It is calculated by applying the softmax function on the log prior probability <math>b_{ij}</math>. The mathematical transformation is shown below (Equation 3 in paper): <br />
<br />
\begin{align}<br />
c_{ij} = \frac{exp(b_ij)}{\sum_{k}exp(b_ik)}<br />
\end{align}<br />
<br />
<math>c_{ij}</math> are served as weights for computing the weighted sum and probabilities. Therefore, as probabilities, they have the following properties:<br />
<br />
\begin{align}<br />
c_{ij} \geq 0, \forall i, j<br />
\end{align}<br />
<br />
and, <br />
<br />
\begin{align}<br />
\sum_{i,j}c_{ij} = 1, \forall i, j<br />
\end{align}<br />
<br />
=== Predicted Output from Layer Below === <br />
<br />
<math>u_{i}</math> are the output vector from capsule i in the lower layer, and <math>\hat{u}_{j|i}</math> are the input vector for capsule j, which are the "prediction vectors" from the capsules in the layer below. <math>\hat{u}_{j|i}</math> is produced by multiplying <math>u_{i}</math> by a weight matrix <math>W_{ij}</math>, such as the following:<br />
<br />
\begin{align}<br />
\hat{u}_{j|i} = W_{ij}u_i<br />
\end{align}<br />
<br />
where <math>W_{ij}</math> encodes some spatial relationship between capsule j and capsule i.<br />
<br />
=== Capsule ===<br />
<br />
By using the definitions from previous sections, the total input vector for an arbitrary capsule j can be defined as:<br />
<br />
\begin{align}<br />
s_j = \sum_{i}c_{ij}\hat{u}_{j|i}<br />
\end{align}<br />
<br />
which is a weighted sum over all prediction vectors by using coupling coefficients.<br />
<br />
=== Squashing ===<br />
<br />
The length of <math>s_j</math> is arbitrary, which is needed to be addressed with. The next step is to convert its length between 0 and 1, since we want the length of the output vector of a capsule to represent the probability that the entity represented by the capsule is present in the current input. The "squashing" process is shown below:<br />
<br />
\begin{align}<br />
v_j = \frac{||s_j||^2}{1+||s_j||^2}\frac{s_j}{||s_j||}<br />
\end{align}<br />
<br />
Notice that "squashing" is not just normalizing the vector into unit length. In addition, it does extra non-linear transformation to ensure that short vectors get shrunk to almost zero length and long vectors get shrunk to a length slightly below 1. The reason for doing this is to make decision of routing, which is called "routing by agreement" much easier to make between capsule layers.<br />
<br />
=== Agreement ===<br />
<br />
The final step of a routing iteration is to form an routing agreement <math>a_{ij}</math>, which is represents as a scalar product:<br />
<br />
\begin{align}<br />
a_{ij} = v_{j}\hat{u}_{j|i}<br />
\end{align}<br />
<br />
As we mentioned in "squashing" section, the length of <math>v_{j}</math> is either close to 0 or close to 1, which will effect the magnitude of <math>a_{ij}</math> in this case. Therefore, the magnitude of <math>a_{ij}</math> indicate the how strong the routing algorithm agrees on taking the route between capsule j and capsule i. For each routing iteration, the log prior probability, <math>b_{ij}</math> will be updated by adding the value of its agreement value, which will effect how the coupling coefficients are computed in the next routing iteration. Because of the "squashing" process, we will eventually end up with a capsule j with its <math>v_{j}</math> close to 1 while all other capsules with its <math>v_{j}</math> close to 0, which indicates that this capsule j should be activated.<br />
<br />
= CapsNet Architecture =<br />
<br />
The second part of this paper discuss the experiment results from a 3-layer CapsNet, the architecture can be divided into two parts, encoder and decoder. <br />
<br />
== Encoder == <br />
<br />
[[File:DRBC_Architecture.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
=== How many routing iteration to use? === <br />
In appendix A of this paper, the authors have shown the empirical results from 500 epochs of training at different choice of routing iterations. According to their observation, more routing iterations increases the capacity of CapsNet but tends to bring additional risk of overfitting. Moreover, CapsNet with routing iterations less than three are not effective in general. As result, they suggest 3 iterations of routing for all experiments.<br />
<br />
=== Marginal loss for digit existence ===<br />
<br />
The experiments performed include segmenting overlapping digits on MultiMINST data set, so the loss function has be adjusted for presents of multiple digits. The marginal lose <math>L_k</math> for each capsule k is calculate by:<br />
<br />
\begin{align}<br />
L_k = T_k max(0, m^+ - ||v_k||)^2 + \lambda(1 - T_k) max(0, ||v_k|| - m^-)^2<br />
\end{align}<br />
<br />
where <math>m^+ = 0.9</math>, <math>m^- = 0.1</math>, and <math>\lambda = 0.5</math>.<br />
<br />
<math>T_k</math> is an indicator for presence of digit of class k, it takes value of 1 if and only if class k is presented. If class k is not presented, <math>\lambda</math> down-weight the loss which shrinks the lengths of the activity vectors for all the digit capsules. By doing this, The loss function penalizes the initial learning for all absent digit class, since we would like the top-level capsule for digit class k to have long instantiation vector if and only if that digit class is present in the input.<br />
<br />
=== Layer 1: Conv1 === <br />
<br />
The first layer of CapsNet. Similar to CNN, this is just convolutional layer that converts pixel intensities to activities of local feature detectors. <br />
<br />
* Layer Type: Convolutional Layer.<br />
* Input: <math>28 \times 28</math> pixels.<br />
* Kernel size: <math>9 \times 9</math>.<br />
* Number of Kernels: 256.<br />
* Activation function: ReLU.<br />
* Output: <math>20 \times 20 \times 256</math> tensor.<br />
<br />
=== Layer 2: PrimaryCapsules ===<br />
<br />
The second layer is formed by 32 primary 8D capsules. By 8D, it means that each primary capsule contains 8 convolutional units with a <math>9 \times 9</math> kernel and a stride of 2. Each capsule will take a <math>20 \times 20 \times 256</math> tensor from Conv1 and produce an output of a <math>6 \times 6 \times 8</math> tensor.<br />
<br />
* Layer Type: Convolutional Layer<br />
* Input: <math>20 \times 20 \times 256</math> tensor.<br />
* Number of capsules: 32.<br />
* Number of convolutional units in each capsule: 8.<br />
* Size of each convolutional unit: <math>6 \times 6</math>.<br />
* Output: <math>6 \times 6 \times 8</math> 8-dimensional vectors.<br />
<br />
=== Layer 3: DigitsCaps ===<br />
<br />
The last layer has 10 16D capsules, one for each digit. Not like the PrimaryCapsules layer, this layer is fully connected. Since this is the top capsule layer, dynamic routing mechanism will be applied between DigitsCaps and PrimaryCapsules. The process begins by taking a transformation of predicted output from PrimaryCapsules layer. Each output is a 8-dimensional vector, which needed to be mapped to a 16-dimensional space. Therefore, the weight matrix, <math>W_{ij}</math> is a <math>8 \times 16</math> matrix. The next step is to acquire coupling coefficients from routing algorithm and to perform "squashing" to get the output. <br />
<br />
* Layer Type: Fully connected layer.<br />
* Input: <math>6 \times 6 \times 8</math> 8-dimensional vectors.<br />
* Output: <math>16 \times 10 </math> matrix.<br />
<br />
=== The loss function ===<br />
<br />
The output of the loss function would be a ten-dimensional one-hot encoded vector with 9 zeros and 1 one at the correct position.<br />
<br />
<br />
== Regularization Method: Reconstruction ==<br />
<br />
This is regularization method introduced in the implementation of CapsNet. The method is to introduce a reconstruction loss (scaled down by 0.0005) to margin loss during training. The authors argue this would encourage the digit capsules to encode the instantiation parameters the input digits. All the reconstruction during training is by using the true labels of the image input. The results from experiments also confirms that adding the reconstruction regularizer enforces the pose encoding in CapsNet and thus boots the performance of routing procedure. <br />
<br />
=== Decoder ===<br />
<br />
The decoder consists of 3 fully connected layers, each layer maps pixel intensities to pixel intensities. The number of parameters in each layer and the activation functions used are indicated in the figure below:<br />
<br />
[[File:DRBC_Decoder.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
=== Result ===<br />
<br />
The authors includes some results for CapsNet classification test accuracy to justify the result of reconstruction. We can see that for CapsNet with 1 routing iteration and CapsNet with 3 routing iterations, implement reconstruction shows significant improvements in both MINIST and MultiMINST data set. These improvements show the importance of routing and reconstruction regularizer. <br />
<br />
[[File:DRBC_Reconstruction.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
= Experiment Results for CapsNet = <br />
<br />
In this part, the authors demonstrate experiment results of CapsNet on different data sets, such as MINIST and different variation of MINST, such as expanded MINST, affNIST, MultiMNIST. Moreover, they also briefly discuss the performance on some other popular data set such CIFAR 10. <br />
<br />
== MINST ==<br />
<br />
=== Highlights ===<br />
<br />
* CapsNet archives state-of-the-art performance on MINST.<br />
* CapsNet with shallow structure (3 layers) achieves performance that only achieves by deeper network before.<br />
<br />
=== Interpretation of Each Capsule ===<br />
<br />
The authors suggest that they found evidence that dimension of some capsule always captures some variance of the digit, while some others represents the global combinations of different variations, this would open some possibility for interpretation of capsules in the future. Some results from perturbations are shown below: <br />
<br />
[[File:DRBC_Dimension.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
== affNIST == <br />
<br />
affNIT data set contains different affine transformation of original MINST data set. By the concept of capsule, CapsNet should gain more robustness from its equivariance nature, and the result confirms this. Compare the baseline CNN, CapsNet achieves 13% improvement on accuracy.<br />
<br />
== MultiMNIST ==<br />
<br />
The MultiMNIST is basically the overlapped version of MINIST. An important point to notice here is that this data set is generated by overlaying a digit on top of another digit from the same set but different class. In other words, the case of stacking digits from the same class is not allowed in MultiMINST. For example, stacking a 5 on a 0 is allowed, but stacking a 5 on another 5 is not. The reason is that CapsNet suffers from the "crowding" effect which will be discussed in the weakness of CapsNet section. <br />
<br />
== Other data sets ==<br />
<br />
CapsNet is used on other data sets such as CIFAR1-, smallNORB and SVHN. The results are not comparable with state-of-the-art performance, but it is still promising since this architecture is the very first, while other networks have been development for a long time.<br />
<br />
= Conclusion = <br />
<br />
This paper discuss the specific part of capsule network, which is the routing-by-agreement mechanism. The authors suggest this is a great approach to solve the current problem with max-pooling in convolutional neural network. Moreover, as author mentioned, the approach mentioned in this paper is only one possible implementation of the capsule concept. The preliminary results from experiment using a simple shallow CapsNet also demonstrate unparalleled performance that indicates the capsules are a direction worth exploring. <br />
<br />
= Weakness of Capsule Network =<br />
<br />
* Routing algorithm introduces internal loops for each capsule. As number of capsules and layers increases, these internal loops may exponentially expand the training time. <br />
* Capsule network suffers a perceptual phenomenon called "crowding", which is common for human vision as well. To address this weakness, capsules have to make a very strong representation assumption that at each location of the image, there is at most one instance of the type of entity that capsule represents. This is also the reason for not allowing overlaying digits from same class in generating process of MultiMINST.<br />
* Other criticisms include that the design of capsule networks requires domain knowledge or feature engineering, contrary to the abstraction-oriented goals of deep learning.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=PointNet%2B%2B:_Deep_Hierarchical_Feature_Learning_on_Point_Sets_in_a_Metric_Space&diff=36054PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space2018-04-03T05:28:22Z<p>J3ku: Add critique about unordered point clouds</p>
<hr />
<div>= Introduction =<br />
This paper builds off of ideas from PointNet (Qi et al., 2017). The name PointNet is derived from the network's input - a point cloud. A point cloud is a set of three dimensional points that each have coordinates <math> (x,y,z) </math>. These coordinates usually represent the surface of an object. For example, a point cloud describing the shape of a torus is shown below.<br />
<br />
[[File:Point cloud torus.gif|thumb|center|Point cloud torus]]<br />
<br />
<br />
Processing point clouds is important in applications such as autonomous driving where point clouds are collected from an onboard LiDAR sensor. These point clouds can then be used for object detection. However, point clouds are challenging to process because:<br />
<br />
# They are unordered. If <math> N </math> is the number of points in a point cloud, then there are <math> N! </math> permutations that the point cloud can be represented.<br />
# The spatial arrangement of the points contains useful information, thus it needs to be encoded.<br />
# The function processing the point cloud needs to be invariant to transformations such as rotation and translations of all points. <br />
<br />
Previously, typical point cloud processing methods handled the challenges of point clouds by transforming the data with a 3D voxel grid or by representing the point cloud with multiple 2D images. When PointNet was introduced, it was novel because it directly took points as its input. PointNet++ improves on PointNet by using a hierarchical method to better capture local structures of the point cloud. <br />
<br />
[[File:point_cloud.png | 400px|thumb|center|Examples of point clouds and their associated task. Classification (left), part segmentation (centre), scene segmentation (right) ]]<br />
<br />
= Related Work =<br />
<br />
This paper provides a new network to understand point clouds. CNNs are the most prominent deep networks which find features in images/videos. However, the convolution process is not applicable to point clouds as point clouds contain unordered set of points with distance metric. Some techniques apply deep learning to unordered sets however they don't account for the distance metric in their model and are sensitive to translation and global normalization. Techniques like volumetric grids and geometric graphs work on the 3D metric space but the problem of non-uniform sampling density hasn't been considered.<br />
<br />
= Review of PointNet =<br />
<br />
The PointNet architecture is shown below. The input of the network is <math> n </math> points, which each have <math> (x,y,z) </math> coordinates. Each point is processed individually through a multi-layer perceptron (MLP). This network creates an encoding for each point; in the diagram, each point is represented by a 1024 dimension vector. Then, using a max pool layer a vector is created that represents the "global signature" of a point cloud. If classification is the task, this global signature is processed by another MLP to compute the classification scores. If segmentation is the task, this global signature is appended to to each point from the "nx64" layer, and these points are processed by a MLP to compute a semantic category score for each point.<br />
<br />
The core idea of the network is to learn a symmetric function on transformed points. Through the T-Nets and the MLP network, a transformation is learned with the hopes of making points invariant to point cloud transformations. Learning a symmetric function solves the challenge imposed by having unordered points; a symmetric function will produce the same value no matter the order of the input. This symmetric function is represented by the max pool layer.<br />
<br />
[[File:pointnet_arch.png | 700px|thumb|center|PointNet architecture. The blue highlighted region is when it is used for classification, and the beige highlighted region is when it is used for segmentation.]]<br />
<br />
= PointNet++ =<br />
<br />
The motivation for PointNet++ is that PointNet does not capture local, fine-grained details. Since PointNet performs a max pool layer over all of its points, information such as the local interaction between points is lost.<br />
<br />
== Problem Statement ==<br />
<br />
There is a metric space <math> X = (M,d) </math> where <math>d</math> is the metric from a Euclidean space <math>\pmb{\mathbb{R}}^n</math> and <math> M \subseteq \pmb{\mathbb{R}}^n </math> is the set of points. The goal is to learn functions <math>f</math> that take <math>X</math> as the input and produce information of semantic interest about it. In practice, <math>f</math> can often be a classification function that outputs a class label or a segmentation function that outputs a per point label for each member of <math>M</math>.<br />
<br />
== Method ==<br />
<br />
=== High Level Overview ===<br />
[[File:point_net++.png | 700px|thumb|right|PointNet++ architecture]]<br />
<br />
The PointNet++ architecture is shown on the right. The core idea is that a hierarchical architecture is used and at each level of the hierarchy a set of points is processed and abstracted to a new set with less points, i.e.,<br />
<br />
\begin{aligned}<br />
\text{Input at each level: } N \times (d + c) \text{ matrix}<br />
\end{aligned}<br />
<br />
where <math>N</math> is the number of points, <math>d</math> is the coordinate points <math>(x,y,z)</math> and <math>c</math> is the feature representation of each point, and<br />
<br />
\begin{aligned}<br />
\text{Output at each level: } N' \times (d + c') \text{ matrix}<br />
\end{aligned}<br />
<br />
where <math>N'</math> is the new number (smaller) of points and <math>c'</math> is the new feature vector.<br />
<br />
<br />
Each level has three layers: Sampling, Grouping, and PointNet. The Sampling layer selects points that will act as centroids of local regions within the point cloud. The Grouping layer then finds points near these centroids. Lastly, the PointNet layer performs PointNet on each group to encode local information.<br />
<br />
=== Sampling Layer ===<br />
<br />
The input of this layer is a set of points <math>{\{x_1,x_2,...,x_n}\}</math>. The goal of this layer is to select a subset of these points <math>{\{\hat{x}_1, \hat{x}_2,...,\hat{x}_m\}} </math> that will define the centroid of local regions.<br />
<br />
To select these points farthest point sampling is used. This is where <math>\hat{x}_j</math> is the most distant point with regards to <math>{\{\hat{x}_1, \hat{x}_2,...,\hat{x}_{j-1}\}}</math>. This ensures coverage of the entire point cloud opposed to random sampling.<br />
<br />
=== Grouping Layer ===<br />
<br />
The objective of the grouping layer is to form local regions around each centroid by grouping points near the selected centroids. The input is a point set of size <math>N \times (d + c)</math> and the coordinates of the centroids <math>N' \times d</math>. The output is the groups of points within each region <math>N' \times k \times (d+c)</math> where <math>k</math> is the number of points in each region.<br />
<br />
Note that <math>k</math> can vary per group. Later, the PointNet layer creates a feature vector that is the same size for all regions at a hierarchical level.<br />
<br />
To determine which points belong to a group a ball query is used; all points within a radius of the centroid are grouped. This is advantageous over nearest neighbour because it guarantees a fixed region space, which is important when learning local structure.<br />
<br />
=== PointNet Layer ===<br />
<br />
After grouping, PointNet is applied to the points. However, first the coordinates of points in a local region are converted to a local coordinate frame by <math> x_i = x_i - \bar{x}</math> where <math>\bar{x}</math> is the coordinates of the centroid.<br />
<br />
=== Robust Feature Learning under Non-Uniform Sampling Density ===<br />
<br />
The previous description of grouping uses a single scale. This is not optimal because the density varies per section of the point cloud. At each level, it would be better if the PointNet layer was applied to adaptively sized groups depending on the point cloud density.<br />
<br />
The two grouping methods the authors propose are shown in the diagram below. Multi-scale grouping (MSG) applies PointNet at various scales per group. The features from the various scales are concatenated to form a multi-scale feature. To train the network to learn an optimal strategy for combining the multi-scale features, the authors proposed random input dropout, which involves randomly dropping input points with a random probability for each training point set. Each input point has a dropout probability <math>\theta</math>. The authors used a <math>\theta</math> value of 0.95. As shown in the experiments section below, dropout provides robustness to input point density variations. During testing stage all points are used. MSG, however, is computationally expensive because for each region it always applies PointNet at large scale neighborhoods to all points. <br />
<br />
On the other hand, multi-resolution grouping (MRG) is less computationally expensive but still adaptively collects features. As shown in the diagram, features of a region from a certain level is a concatenation of two vectors. The left vector is obtained by applying PointNet to three points, and these three points obtained information from three groups. This vector is then concatenated by a vector that is created by using PointNet on all the points in the level below. The second vector can be weighed more heavily if the first vector contains a sparse amount of points, since the first vector is based on subregions that would be even more sparse and suffer from sampling deficiency. On the other hand, when the density of a local region is high, the first vector can be weighted more heavily as it allows for inspecting at higher resolutions in the lower levels to obtain finer details. <br />
<br />
[[File:grouping.png | 300px|thumb|center|Example of the two ways to perform grouping]]<br />
<br />
== Point Cloud Segmentation ==<br />
<br />
If the task is segmentation, the architecture is slightly modified since we want a semantic score for each point. To achieve this, distance-based interpolation and skip-connections are used.<br />
<br />
=== Distance-based Interpolation ===<br />
<br />
Here, point features from <math>N_l \times (d + C)</math> points are propagated to <math>N_{l-1} \times (d + C)</math> points where <math>N_{l-1}</math> is greater than <math>N_l</math>.<br />
<br />
To propagate features an inverse distance weighted average based on <math>k</math> nearest neighbors is used. The <math>p=2</math> and <math>k=3</math>.<br />
<br />
[[File:prop_feature.png | 500px|thumb|center|Feature interpolation during segmentation]]<br />
<br />
=== Skip-connections ===<br />
<br />
In addition, skip connections are used (see the PointNet++ architecture diagram). The features from the the skip layers are concatenated with the interpolated features. Next, a "unit-wise" PointNet is applied, which the authors describe as similar to a one-by-one convolution.<br />
<br />
== Experiments ==<br />
To validate the effectiveness of PointNet++, experiments in three areas were performed - classification in Euclidean metric space, semantic scene labelling, and classification in non-Euclidean space.<br />
<br />
=== Point Set Classification in Euclidean Metric Space ===<br />
<br />
The digit dataset, MNIST, was converted to a 2D point cloud. Pixel intensities were normalized in the range of <math>[0, 1]</math>, and only pixels with intensities larger than 0.5 were considered. The coordinate system was set at the centre of the image. PointNet++ achieved a classification error of 0.51%. The original PointNet had 0.78% classification error. The table below compares these results to the state-of-the-art.<br />
<br />
[[File:mnist_results.png | 300px|thumb|center|MNIST classification results.]]<br />
<br />
In addition, the ModelNet40 dataset was used. This dataset consists of CAD models. Three dimensional point clouds were sampled from mesh surfaces of the ModelNet40 shapes. The classification results from this dataset are shown below. The last row in the table below, "Ours (with normal)" used face normals (normal is the same for the entire face, regardless of the point picked on that face) as additional point features as well as additional points <math>(N = 5000)</math> to boost performance. All these points are normalized to have zero mean and be within one unit ball. The network contains three hierarchical levels with three fully connected layers.<br />
<br />
[[File:modelnet40.png | 300px|thumb|center|ModelNet40 classification results.]]<br />
<br />
An experiment was performed to show how the accuracy was affected by the number of points used. With PointNet++ using multi-scale grouping and dropout, the performance decreased by less than 1% when 1024 test points were reduced to 256. On the other hand, PointNet's performance was impacted by the decrease in points. This is not surprising because the dropout feature of PointNet++ ensures that the model is trained specifically to be robust to loss of points. <br />
<br />
[[File:paper28_fig4_chair.png | 300px|thumb|center|An example showing the reduction of points visually. At 256 points, the points making up the object is very spare, however the accuracy is only reduced by 1%]][[File:num_points_acc.png | 300px|thumb|center|Relationship between accuracy and the number of points used for classification.]]<br />
<br />
=== Semantic Scene Labelling ===<br />
<br />
The ScanNet dataset was used for experiments in semantic scene labelling. This dataset consists of laser scans of indoor scenes where the goal is to predict a semantic label for each point. Example results are shown below.<br />
<br />
[[File:scannet.png | 300px|thumb|center|Example ScanNet semantic segmentation results.]]<br />
<br />
To compare to other methods, the authors convert their point labels to a voxel format, and accuracy is determined on a per voxel basis. The accuracy compared to other methods is shown below.<br />
<br />
[[File:scannet_acc.png | 500px|thumb|center|ScanNet semantic segmentation classification comparison to other methods.]]<br />
<br />
To test how the trained model performed on scans with non-uniform sampling density, virtual scans of Scannet scenes were synthesized and the network was evaluated on this data. It can be seen from the above figures that SSG performance greatly falls due to the sampling density shift. MRG network, on the other hand, is more robust to the sampling density shift since it is able to automatically switch to features depicting coarser granularity when the sampling is sparse. This proves the effectiveness of the proposed density adaptive layer design.<br />
<br />
=== Classification in Non-Euclidean Metric Space ===<br />
<br />
[[File:shrec.png | 300px|thumb|right|Example of shapes from the SHREC15 dataset.]]<br />
<br />
Lastly, experiments were performed on the SHREC15 dataset. This dataset contains shapes that have different poses. This experiment shows that PointNet++ is able to generalize to non-Euclidean spaces. Results from this dataset are provided below.<br />
<br />
[[File:shrec15_results.png | 500px|thumb|center|Results from the SHREC15 dataset.]]<br />
<br />
=== Feature Visualization ===<br />
The figure below visualizes what is learned by just the first layer kernels of the network. The model is trained on a dataset the mostly consisted of furniture which explains the lines, corners, and planes visible in the visualization. Visualization is performed by creating a voxel grid in space and only aggregating point sets that activate specific neurons the most.<br />
<br />
[[File:26_8.PNG | 800px|thumb|center|Pointclouds learned from first layer kernels (red is near, blue is far)]]<br />
<br />
=== Time and Space Complexity ===<br />
To evaluate the time and space complexity between PointNet++ and PointNet the authors recorded the model size and inference time for several point set deep learning approaches. Inference time is measured with a GTX 1080 and batch size 8. The PointNet inference times are significantly better however the model sizes for PointNet++ are comparable. The table below outlines the specifics for each method.<br />
<br />
Worth noting is that ours MSG, while it has good performance in non-uniformly sampled data, it’s twice as expensive as then SSG version due the multi-scale region feature extraction. Compared with MSG,<br />
MRG is more efficient since it uses regions across layers.<br />
<br />
[[File:pointnet_complexity.PNG | 700px|thumb|center|Comparison of model size and inference time between PointNet and PointNet++]]<br />
<br />
== Critique ==<br />
<br />
It seems clear that PointNet is lacking capturing local context between points. PointNet++ seems to be an important extension, but the improvements in the experimental results seem small. Some computational efficiency experiments would have been nice. For example, the processing speed of the network, and the computational efficiency of MRG over MRG.<br />
<br />
It may be useful to note that not all raw point clouds coming from sensors are completely unordered. For example, the points from a LiDAR scanner are ordered by the specific angles of the laser scans, which can be seen as rings in the point cloud (shown in the figure below). The discontinuities along each ring could be used to provide trackable feature information for SLAM algorithms, and would be harder to recover if the point cloud is represented in an unordered manner.<br />
<br />
[[File:point_net_pp_lidar_scan.png | 500px|thumb|center|Example LiDAR point cloud]]<br />
<br />
== Code ==<br />
<br />
Code for PointNet++ can be found at: https://github.com/charlesq34/pointnet2 <br />
<br />
<br />
=Sources=<br />
1. Charles R. Qi, Li Yi, Hao Su, Leonidas J. Guibas. PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space, 2017<br />
<br />
2. Charles R. Qi, Hao Su, Kaichun Mo, Leonidas J. Guibas. PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation, 2017</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:point_net_pp_lidar_scan.png&diff=36053File:point net pp lidar scan.png2018-04-03T05:25:32Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Neural_Audio_Synthesis_of_Musical_Notes_with_WaveNet_autoencoders&diff=36052Neural Audio Synthesis of Musical Notes with WaveNet autoencoders2018-04-03T05:05:01Z<p>J3ku: Add baseline spectral autoencoder experiments</p>
<hr />
<div>= Introduction =<br />
The authors of this paper have pointed out that the method in which most notes are created are hand-designed instruments modifying pitch, velocity and filter parameters to produce the required tone, timbre and dynamics of a sound. The authors suggest that this may be a problem and thus suggest a data-driven approach to audio synthesis. They demonstrate how to generate new types of expressive and realistic instrument sounds using a neural network model instead of using specific arrangements of oscillators or algorithms for sample playback. The model is capable of learning semantically meaningful hidden representations which can be used as control signals for manipulating tone, timbre, and dynamics during playback. To train such a data expensive model the authors highlight the need for a large dataset much like ImageNet for music. The motivation for this work stems from recent advances in autoregressive models like WaveNet [5] and SampleRNN [6]. These models are effective at modeling short and medium scale (~500ms) signals, but rely on external conditioning for large-term dependencies; the proposed model removes the need for external conditioning.<br />
<br />
= Contributions =<br />
To solve the problem highlighted above the authors propose two main contributions of their paper: <br />
* Wavenet-style autoencoder that learn to encode temural data over a long term audio structures without requiring external conditioning<br />
* NSynth: a large dataset of musical notes inspired by the emerging of large image datasets<br />
<br />
<br />
= Models =<br />
<br />
[[File:paper26-figure1-models.png|center]]<br />
<br />
== WaveNet Autoencoder ==<br />
<br />
While the proposed autoencoder structure is very similar to that of WaveNet the authors argue that the algorithm is novel in two ways:<br />
* It is able to attain consistent long-term structure without any external conditioning <br />
* Creating meaningful embedding which can be interpolated between<br />
The authors accomplish this by passing the raw audio throw the encoder to produce an embedding <math>Z = f(x) </math>, next the input is shifted and feed into the decoder which reproduces the input. The resulting probability distribution: <br />
<br />
\begin{align}<br />
p(x) = \prod_{i=1}^N\{x_i | x_1, … , x_N-1, f(x) \}<br />
\end{align}<br />
<br />
A detailed block diagram of the modified WaveNet structure can be seen in figure 1b. This diagram demonstrates the encoder as a 30 layer network in each each node is a ReLU nonlinearity followed by a non-causal dilated convolution. Dilated convolution (aka convolutions with holes) is a type of convolution in which the filter skips input values with a certain step (step size of 1 is equivalent to the standard convolution), effectively allowing the network to operate at a coarser scale compared to traditional convolutional layers and have very large receptive fields. The resulting convolution is 128 channels all feed into another ReLU nonlinearity which is feed into another 1x1 convolution before getting down sampled with average pooling to produce a 16 dimension <math>Z </math> distribution. Each <math>Z </math> encoding is for a specific temporal resolution which the authors of the paper tuned to 32ms. This means that there are 125, 16 dimension <math>Z </math> encodings for each 4 second note present in the NSynth database (1984 embeddings). <br />
Before the <math>Z </math> embedding enters the decoder it is first upsampled to the original audio rate using nearest neighbor interpolation. The embedding then passes through the decoder to recreate the original audio note. The input audio data is first quantized using 8-bit mu-law encoding into 256 possible values, and the output prediction is the softmax over the possible values.<br />
<br />
== Baseline: Spectral Autoencoder ==<br />
Being unable to find an alternative fully deep model which the authors could use to compare to there proposed WaveNet autoencoder to, the authors just made a strong baseline. The baseline algorithm that the authors developed is a spectral autoencoder. The block diagram of its architecture can be seen in figure 1a. The baseline network is 10 layer deep. Each layer has a 4x4 kernels with 2x2 strides followed by a leaky-ReLU (0.1) and batch normalization. The final hidden vector(Z) was set to 1984 to exactly match the hidden vector of the WaveNet autoencoder. <br />
<br />
Given the simple architecture, the authors first attempted to train the baseline on raw waveforms as input, with a mean-squared error cost. This did not work well and showed the problem of the independent Gaussian assumption. Spectral representations from FFT worked better, but had low perceptual quality despite having low MSE cost after training. Training on the log magnitude of the power spectra, normalized between 0 and 1, was found to be best correlated with perceptual distortion. The authors also explored several representations of phase, finding that estimating magnitude and using established iterative techniques to reconstruct phase to be most effective. (The technique to reconstruct the phase from the magnitude comes from (Griffin and Lim 1984). It can be summarized as follows. In each iteration, generate a Fourier signal z by taking the Short Time Fourier transform of the current estimate of the complete time-domain signal, and replacing its magnitude component with the known true magnitude. Then find the time-domain signal whose Short Time Fourier transform is closest to z in the least-squares sense. This is the estimate of the complete signal for the next iteration. ) A final heuristic that was used by the authors to increase the accuracy of the baseline was weighting the mean square error (MSE) loss starting at 10 for 0 HZ and decreasing linearly to 1 at 4000 Hz and above. This is valid as the fundamental frequency of most instrument are found at lower frequencies. <br />
<br />
== Training ==<br />
Both the modified WaveNet and the baseline autoencoder used stochastic gradient descent with an Adam optimizer. The authors trained the baseline autoencoder model asynchronously for 1800000 epocs with a batch size of 8 with a learning rate of 1e-4. Where as the WaveNet modules were trained synchronously for 250000 epocs with a batch size of 32 with a decaying learning rate ranging from 2e-4 to 6e-6.<br />
<br />
= The NSynth Dataset =<br />
To evaluate the WaveNet autoencoder model, the authors' wanted an audio dataset that let them explore the learned embeddings. Musical notes are an ideal setting for this study. Prior to this paper, the existing music datasets included the RWC music database (Goto et al., 2003) and the dataset from Romani Picas et al. However, the authors wanted to develop a larger dataset.<br />
<br />
The NSynth dataset has 306 043 unique musical notes (each have a unique pitch, timbre, envelope) all 4 seconds in length sampled at 16,000 Hz. The data set consists of 1006 different instruments playing on average of 65.4 different pitches across on average 4.75 different velocities. Average pitches and velocities are used as not all instruments, can reach all 88 MIDI frequencies, or the 5 velocities desired by the authors. The dataset has the following split: training set with 289,205 notes, validation set with 12,678 notes, and test set with 4,096 notes.<br />
<br />
Along with each note the authors also included the following annotations:<br />
* Source - The way each sound was produced. There were 3 classes ‘acoustic’, ‘electronic’ and ‘synthetic’.<br />
* Family - The family class of instruments that produced each note. There are 11 classes which include: {‘bass’, ‘brass’, ‘vocal’ ext.}<br />
* Qualities - Sonic qualities about each note<br />
<br />
The full dataset is publicly available here: https://magenta.tensorflow.org/datasets/nsynth as TFRecord files with training and holdout splits.<br />
<br />
[[File:nsynth_table.png | 400px|thumb|center|Full details of the NSynth dataset.]]<br />
<br />
= Evaluation =<br />
<br />
To fully analyze all aspects of WaveNet the authors proposed three evaluations:<br />
* Reconstruction - Both Quantitative and Qualitative analysis were considered<br />
* Interpolation in Timbre and Dynamics<br />
* Entanglement of Pitch and Timbre <br />
<br />
Sound is historically very difficult to quantify from a picture representation as it requires training and expertise to analyze. Even with expertise it can be difficult to complete a full analysis as two very different sounds can look quite similar in their respective pictorial representations. This is why the authors recommend all readers to listen to the created notes which can be found here: https://magenta.tensorflow.org/nsynth.<br />
<br />
However, even when taking this under consideration the authors do pictorially demonstrate differences in the two proposed algorithms along with the original note, as it is hard to publish a paper with sound included. To demonstrate the pictorial difference the authors demonstrate each note using constant-q transform (CQT) which is able to capture the dynamics of timbre along with representing the frequencies of the sound.<br />
<br />
== Reconstruction ==<br />
<br />
[[File:paper27-figure2-reconstruction.png|center]]<br />
<br />
=== Qualitative Comparison ===<br />
In the Glockenspiel the WaveNet autoencoder is able to reproduce the magnitude, phase of the fundamental frequency (A and C in figure 2), and the attack (B in figure 2) of the instrument; Whereas the Baseline autoencoder introduces non existing harmonics (D in figure 2). The flugelhorn on the other hand, presents the starkest difference between the WaveNet and baseline autoencoders. The WaveNet while not perfect is able to reproduce the verbarto (I and J in figure 2) across multiple frequencies, which results in a natural sounding note. The baseline not only fails to do this but also adds extra noise (K in figure 2). The authors do add that the WaveNet produces some strikes (L in figure 2) however they argue that they are inaudible.<br />
<br />
[[File:paper27-table1.png|center]]<br />
<br />
Mu-law encoding was used in the original WaveNet [https://arxiv.org/pdf/1609.03499.pdf paper] to make the problem "more tractable" compared to raw 16-bit integer values. In that paper, they note that "especially for speech, this non-linear quantization produces a significantly better reconstruction" compared to a linear scheme. This might be expected considering that the mu-law companding transformation was designed to [https://www.cisco.com/c/en/us/support/docs/voice/h323/8123-waveform-coding.html#t4 encode speech]. In this application though, using this encoding creates perceptible distortion that sounds similar to clipping.<br />
<br />
=== Quantitative Comparison ===<br />
For a quantitative comparison the authors trained a separate multi-task classifier to classify a note using given pitch or quality of a note. The results of both the Baseline and the WaveNet where then inputted and attempted to be classified. As seen in table 1 WaveNet significantly outperformed the Baseline in both metrics posting a ~70% increase when only considering pitch.<br />
<br />
== Interpolation in Timbre and Dynamics ==<br />
<br />
[[File:paper27-figure3-interpolation.png|center]]<br />
<br />
For this evaluation the authors reconstructed from linear interpolations in Z space among different instruments and compared these to superimposed position of the original two instruments. Not surprisingly the model fuse aspects of both instruments during the recreation. The authors claim however, that WaveNet produces much more realistic sounding results. <br />
To support their claim the authors the authors point to WaveNet ability to create dynamic mixing of overtone in time, even jumping to higher harmonics (A in figure 3), capturing the timbre and dynamics of both the bass and flute. This can be once again seen in (B in figure 3) where Wavenet adds additional harmonics as well as a sub-harmonics to the original flute note. <br />
<br />
<br />
== Entanglement of Pitch and Timbre ==<br />
<br />
[[File:paper27-table2.png|center]]<br />
<br />
[[File:paper27-figure4-entanglement.png|center]]<br />
<br />
To study the entanglement between pitch and Z space the authors constructed a classifier which was expected to drop in accuracy if the representation of pitch and timbre is disentangled as it relies heavily on the pitch information. This is clearly demonstrated by the first two rows of table 2 where WaveNet relies more strongly on pitch then the baseline algorithm. The authors provide a more qualitative demonstrating in figure 4. They demonstrate a situation in which a classifier may be confused; a note with pitch of +12 is almost exactly the same as the original apart from an emergence of sub-harmonics.<br />
<br />
Further insight can be gained on the relationship between pitch and timbre by studying the trend amongst the network embeddings among the pitches for specific instruments. This is depicted in figure 5 for several instruments across their entire 88 note range at 127 velocity. It can be noted from the figure that the instruments have unique separation of two or more registers over which the embeddings of notes with different pitches are similar. This is expected since instrumental dynamics and timbre varies dramatically over the range of the instrument.<br />
<br />
= Conclusion & Future Directions =<br />
<br />
This paper presents a Wavelet autoencoder model which is built on top of the WaveNet model and evaluate the model on NSynth dataset. The paper also introduces a new large scale dataset of musical notes: NSynth.<br />
<br />
One significant area which the authors claim great improvement is needed is the large memory constraints required by there algorithm. Due to the large memory requirement the current WaveNet must rely on down sampling thus being unable to fully capture the global context. This is an area where model compression techniques could be beneficial. That is, quantization and pruning could be effective: with 4-bit quantization during the entire process (weights, activations, gradients, error as in the work of Wu et al., 2016), memory requirement could be reduced by at least 8 times. The authors also claim that research using different input representations (instead of mu-law) to minimize distortion is ongoing.<br />
<br />
= Critique = <br />
* Authors have never conducted a human study determining sound similarity between the original, baseline, and WaveNet.<br />
* Architecture is not very novel<br />
<br />
= Open Source Code =<br />
<br />
Google has released all code related to this paper at the following open source repository: https://github.com/tensorflow/magenta/tree/master/magenta/models/nsynth<br />
<br />
= References =<br />
<br />
# Engel, J., Resnick, C., Roberts, A., Dieleman, S., Norouzi, M., Eck, D. & Simonyan, K.. (2017). Neural Audio Synthesis of Musical Notes with WaveNet Autoencoders. Proceedings of the 34th International Conference on Machine Learning, in PMLR 70:1068-1077<br />
# Griffin, Daniel, and Jae Lim. "Signal estimation from modified short-time Fourier transform." IEEE Transactions on Acoustics, Speech, and Signal Processing 32.2 (1984): 236-243.<br />
# NSynth: Neural Audio Synthesis. (2017, April 06). Retrieved March 19, 2018, from https://magenta.tensorflow.org/nsynth <br />
# The NSynth Dataset. (2017, April 05). Retrieved March 19, 2018, from https://magenta.tensorflow.org/datasets/nsynth<br />
# Oord, Aaron van den, Nal Kalchbrenner, and Koray Kavukcuoglu. "Pixel recurrent neural networks." arXiv preprint arXiv:1601.06759 (2016).<br />
# Mehri, Soroush, et al. "SampleRNN: An unconditional end-to-end neural audio generation model." arXiv preprint arXiv:1612.07837 (2016).<br />
# Wu, S., Li, G., Chen, F., & Shi, L. (2018). Training and Inference with Integers in Deep Neural Networks. arXiv preprint arXiv:1802.04680.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Do_Deep_Neural_Networks_Suffer_from_Crowding&diff=36051Do Deep Neural Networks Suffer from Crowding2018-04-03T04:44:29Z<p>J3ku: Add experiment results of flankers from different datasets</p>
<hr />
<div>= Introduction =<br />
Since the increase in popularity of Deep Neural Networks (DNNs), there has been lots of research in making machines capable of recognizing objects the same way humans do. Humans can recognize objects in a way that is invariant to scale, translation, and clutter. Crowding is visual effect suffered by humans, in which an object that can be recognized in isolation can no longer be recognized when other objects, called flankers, are placed close to it. This paper focuses on studying the impact of crowding on DNNs trained for object recognition by adding clutter to the images and then analyzing which models and settings suffer less from such effects. <br />
<br />
[[File:paper25_fig_crowding_ex.png|center|600px]]<br />
The figure shows a visual example of crowding [3]. Keep your eyes still and look at the dot in the center and try to identify the "A" in the two circles. You should see that it is much easier to make out the "A" in the right than in the left circle. The same "A" exists in both circles, however, the left circle contains flankers which are those line segments.<br />
<br />
Another common example to visualize the same:<br />
[[File:crowding-tigger.jpg|center|600px]]<br />
<br />
<br />
===What is the problem in CNNs?===<br />
CNNs fall short in explaining human perceptual invariance. First, CNNs typically take input at a single uniform resolution. Biological measurements suggest that resolution is not uniform across the human visual field, but rather decays with eccentricity, i.e. distance from the center of focus. Even more importantly, CNNs rely not only on weight-sharing but also on data augmentation to achieve transformation invariance and so obviously a lot of processing is needed for CNNs.<br />
<br />
The paper investigates two types of DNNs for crowding: traditional deep convolutional neural networks (DCNN) and a multi-scale eccentricity-dependent model which is an extension of the DCNNs and inspired by the retina where the receptive field size of the convolutional filters in the model grows with increasing distance from the center of the image, called the eccentricity and is explained below. The authors focus on the dependence of crowding on image factors, such as flanker configuration, target-flanker similarity, target eccentricity and premature pooling in particular. Along with that, there is major emphasis on reducing the training time of the networks since the motive is to have a simple network capable of learning space-invariant features.<br />
<br />
= Models =<br />
The authors describe two kinds of DNN architectures: Deep Convolutional Neural Networks, and eccentricity dependent networks, with varying pooling strategies across space and scale. Of particular note is the pooling operation, as many researchers have suggested that this may be the cause of crowding in human perception.<br />
<br />
== Deep Convolutional Neural Networks ==<br />
The DCNN is a basic architecture with 3 convolutional layers, spatial 3x3 max-pooling with varying strides and a fully connected layer for classification as shown in the below figure. <br />
[[File:DCNN.png|800px|center]]<br />
<br />
The network is fed with images resized to 60x60, with mini-batches of 128 images, 32 feature channels for all convolutional layers, and convolutional filters of size 5x5 and stride 1.<br />
<br />
As highlighted earlier, the effect of pooling is into main consideration and hence three different configurations have been investigated as below: <br />
<br />
# '''No total pooling''' Feature maps sizes decrease only due to boundary effects, as the 3x3 max pooling has stride 1. The square feature maps sizes after each pool layer are 60-54-48-42.<br />
# '''Progressive pooling''' 3x3 pooling with a stride of 2 halves the square size of the feature maps, until we pool over what remains in the final layer, getting rid of any spatial information before the fully connected layer. (60-27-11-1).<br />
# '''At end pooling''' Same as no total pooling, but before the fully connected layer, max-pool over the entire feature map. (60-54-48-1).<br />
<br />
==Eccentricity-dependent Model==<br />
In order to take care of the scale invariance in the input image, the eccentricity dependent DNN is utilized. This was proposed as a model of the human visual cortex by [https://arxiv.org/pdf/1406.1770.pdf, Poggio et al] and later further studied in [2]. The main intuition behind this architecture is that as we increase eccentricity, the receptive fields also increase and hence the model will become invariant to changing input scales. The authors note that the width of each scale is roughly related to the amount of translation invariance for objects at that scale, simply because once the object is outside that window, the filter no longer observes it. Therefore, the authors say that the architecture emphasizes scale invariance over translation invariance, in contrast to traditional DCNNs. From a biological perspective, eye movement can compensate for the limitations of translation invariance, but compensating for scale invariance requires changing distance from the object. In this model, the input image is cropped into varying scales (11 crops increasing by a factor of <math>\sqrt{2}</math> which are then resized to 60x60 pixels) and then fed to the network. Exponentially interpolated crops are used over linearly interpolated crops since they produce fewer boundary effects while maintaining the same behavior qualitatively. The model computes an invariant representation of the input by sampling the inverted pyramid at a discrete set of scales with the same number of filters at each scale. Since the same number of filters are used for each scale, the smaller crops will be sampled at a high resolution while the larger crops will be sampled with a low resolution. These scales are fed into the network as an input channel to the convolutional layers and share the weights across scale and space. Due to the downsampling of the input image, this is equivalent to having receptive fields of varying sizes. Intuitively, this means that the network generalizes learnings across scales and is guaranteed by during back-propagation by averaging the error derivatives over all scale channels, then using the averages to compute weight adjustments. The same set of weight adjustments to the convolutional units across different scale channels is applied.<br />
[[File:EDM.png|2000x450px|center]]<br />
<br />
<br />
The architecture of this model is the same as the previous DCNN model with the only change being the extra filters added for each of the scales, so the number of parameters remains the same as DCNN models. The authors perform spatial pooling, the aforementioned ''At end pooling'' is used here, and scale pooling which helps in reducing the number of scales by taking the maximum value of corresponding locations in the feature maps across multiple scales. It has three configurations: (1) at the beginning, in which all the different scales are pooled together after the first layer, 11-1-1-1-1 (2) progressively, 11-7-5-3-1 and (3) at the end, 11-11-11-11-1, in which all 11 scales are pooled together at the last layer.<br />
<br />
===Contrast Normalization===<br />
Since there are multiple scales of an input image, in some experiments, normalization is performed such that the sum of the pixel intensities in each scale is in the same range [0,1] (this is to prevent smaller crops, which have more non-black pixels, from disproportionately dominating max-pooling across scales). The normalized pixel intensities are then divided by a factor proportional to the crop area [[File:sqrtf.png|60px]] where i=1 is the smallest crop.<br />
<br />
=Experiments=<br />
Targets are the set of objects to be recognized and flankers are the set of objects the model has not been trained to recognize, which act as clutter with respect to these target objects. The target objects are the even MNIST numbers having translational variance (shifted at different locations of the image along the horizontal axis), while flankers are from odd MNIST numbers, not MNIST dataset (contains alphabet letters) and Omniglot dataset (contains characters). Examples of the target and flanker configurations are shown below: <br />
[[File:eximages.png|800px|center]]<br />
<br />
The target and the object are referred to as ''a'' and ''x'' respectively with the below four configurations: <br />
# No flankers. Only the target object. (a in the plots) <br />
# One central flanker closer to the center of the image than the target. (xa) <br />
# One peripheral flanker closer to the boundary of the image that the target. (ax) <br />
# Two flankers spaced equally around the target, being both the same object, see Figure 1 above for an example (xax).<br />
<br />
Training is done using backpropagation with images of size <math>1920 px^2</math> with embedded targets objects and flankers of size of <math>120 px^2</math>. The training and test images are divided as per the usual MNIST configuration. To determine if there is a difference between the peripheral flankers and the central flankers, all the tests are performed in the right half image plane.<br />
<br />
==DNNs trained with Target and Flankers==<br />
This is a constant spacing training setup where identical flankers are placed at a distance of 120 pixels either side of the target(xax) with the target having translational variance. The tests are evaluated on (i) DCNN with at the end pooling, and (ii) eccentricity-dependent model with 11-11-11-11-1 scale pooling, at the end spatial pooling and contrast normalization. The test data has different flanker configurations as described above.<br />
[[File:result1.png|x450px|center]]<br />
<br />
===Observations===<br />
* With the flanker configuration same as the training one, models are better at recognizing objects in clutter rather than isolated objects for all image locations<br />
* If the target-flanker spacing is changed, then models perform worse<br />
* the eccentricity model is much better at recognizing objects in isolation than the DCNN because the multi-scale crops divide the image into discrete regions, letting the model learn from image parts as well as the whole image<br />
* Only the eccentricity-dependent model is robust to different flanker configurations not included in training when the target is centered.<br />
<br />
==DNNs trained with Images with the Target in Isolation==<br />
Here the target objects are in isolation and with translational variance while the test-set is the same set of flanker configurations as used before. The constant spacing and constant eccentricity effect have been evaluated.<br />
<br />
[[File:result2.png|750x400px|center]]<br />
<br />
In addition to the evaluation of DCNNs in constant target eccentricity at 240 pixels, here they are tested with images in which the target is fixed at 720 pixels from the center of the image, as shown in Fig 3. Since the target is already at the edge of the visual field, a flanker cannot be more peripheral in the image than the target. Same results as for the 240 pixels target eccentricity can be extracted. The closer the flanker is to the target, the more accuracy decreases. Also, it can be seen that when the target is close to the image boundary, recognition is poor because of boundary effects eroding away information about the target.<br />
<br />
The authors also test the effect of flankers from different datasets on a DCNN model with at end pooling, with results shown in Fig. 7 below. Omniglot flankers crowd less than MNIST digits, and the authors note that this is because they are visually similar to MNIST digits, but are not actually digits, and thus activate the model's convolutional filters less than MNIST digits.<br />
<br />
[[File:crowding_at_end_pooling.png|750px|center]]<br />
<br />
[[File:paper25_supplemental1.png|800px|center]]<br />
<br />
===DCNN Observations===<br />
* The recognition gets worse with the increase in the number of flankers.<br />
* Convolutional networks are capable of being invariant to translations.<br />
* In the constant target eccentricity setup, where the target is fixed at the center of the image with varying target-flanker spacing, we observe that as the distance between target and flankers increase, recognition gets better.<br />
* Spatial pooling helps the network in learning invariance.<br />
* Flankers similar to the target object helps in recognition since they activate the convolutional filter more.<br />
* notMNIST data affects leads to more crowding since they have many more edges and white image pixels which activate the convolutional layers more.<br />
<br />
===Eccentric Model===<br />
The set-up is the same as explained earlier.<br />
[[File:result3.png|750x400px|center]]<br />
<br />
====Observations====<br />
* The recognition accuracy is dependent on the eccentricity of the target object.<br />
* If the target is placed at the center and no contrast normalization is done, then the recognition accuracy is high since this model concentrates the most on the central region of the image.<br />
* If contrast normalization is done, then all the scales will contribute equal amount and hence the eccentricity dependence is removed.<br />
* Early pooling is harmful since it might take away the useful information very early which might be useful to the network.<br />
<br />
Without contrast normalization, the middle portion of the image can be focused more with high resolution so the target at the center with no normalization performs well in that case. But if normalization is done, then all the segments of the image contribute to the classification and hence the overall accuracy is not that great but the system becomes robust to the changes in eccentricity.<br />
<br />
==Complex Clutter==<br />
Here, the targets are randomly embedded into images of the Places dataset and shifted along horizontally in order to investigate model robustness when the target is not at the image center. Tests are performed on DCNN and the eccentricity model with and without contrast normalization using at end pooling. The results are shown in Figure 9 below. <br />
<br />
[[File:result4.png|750x400px|center]]<br />
<br />
====Observations====<br />
* Only eccentricity model without contrast normalization can recognize the target and only when the target is close to the image center.<br />
* The eccentricity model does not need to be trained on different types of clutter to become robust to those types of clutter, but it needs to fixate on the relevant part of the image to recognize the target. If it can fixate on the relevant part of the image, it can still discriminate it, even at different scales. This implies that the eccentricity model is robust to clutter.<br />
<br />
=Conclusions=<br />
This paper investigates the effect of crowding on a DNN. Using a simple technique of adding clutter in the model didn't improve the performance. We often think that just training the network with data similar to the test data would achieve good results in a general scenario too but that's not the case as we trained the model with flankers and it did not give us the ideal results for the target objects. The following 4 techniques influenced crowding in DNN:<br />
*'''Flanker Configuration''': When models are trained with images of objects in isolation, adding flankers harms recognition. Adding two flankers is the same or worse than adding just one and the smaller the spacing between flanker and target, the more crowding occurs. This is because the pooling operation merges nearby responses, such as the target and flankers if they are close.<br />
*'''Similarity between target and flanker''': Flankers more similar to targets cause more crowding, because of the selectivity property of the learned DNN filters.<br />
*'''Dependence on target location and contrast normalization''': In DCNNs and eccentricity-dependent models with contrast normalization, recognition accuracy is the same across all eccentricities. In eccentricity-dependent networks without contrast normalization, recognition does not decrease despite the presence of clutter when the target is at the center of the image.<br />
*'''Effect of pooling''': adding pooling leads to better recognition accuracy of the models. Yet, in the eccentricity model, pooling across the scales too early in the hierarchy leads to lower accuracy.<br />
* The Eccentricity Dependent Models can be used for modeling the feedforward path of the primate visual cortex. <br />
* If target locations are proposed, then the system can become even more robust and hence a simple network can become robust to clutter while also reducing the amount of training data and time needed<br />
<br />
=Critique=<br />
This paper only tries to check the impact of flankers on targets as to how crowding can affect recognition but it does not propose anything novel in terms of architecture to take care of such type of crowding. The paper only shows that the eccentricity based model does better (than plain DCNN model) when the target is placed at the center of the image but maybe windowing over the frames the same way that a convolutional model passes a filter over an image, instead of taking crops starting from the middle, might help.<br />
<br />
This paper focuses on image classification. For a stronger argument, their model could be applied to the task of object detection. Perhaps crowding does not have as large of an impact when the objects of interest are localized by a region proposal network.<br />
<br />
This paper does not provide a convincing argument that the problem of crowding as experienced by humans somehow shares a similar mechanism to the problem of DNN accuracy falling when there is more clutter in the scene. The multi-scale architecture does not appear similar to the distribution of rods and cones in the retina[https://www.ncbi.nlm.nih.gov/books/NBK10848/figure/A763/?report=objectonly]. It might be that the eccentric model does well when the target is centered because it is being sampled by more scales, not because it is similar to a primate visual cortex, and primates are able to recognize an object in clutter when looking directly at it.<br />
<br />
=References=<br />
# Volokitin A, Roig G, Poggio T:"Do Deep Neural Networks Suffer from Crowding?" Conference on Neural Information Processing Systems (NIPS). 2017<br />
# Francis X. Chen, Gemma Roig, Leyla Isik, Xavier Boix and Tomaso Poggio: "Eccentricity Dependent Deep Neural Networks for Modeling Human Vision" Journal of Vision. 17. 808. 10.1167/17.10.808.<br />
# J Harrison, W & W Remington, R & Mattingley, Jason. (2014). Visual crowding is anisotropic along the horizontal meridian during smooth pursuit. Journal of vision. 14. 10.1167/14.1.21. http://willjharrison.com/2014/01/new-paper-visual-crowding-is-anisotropic-along-the-horizontal-meridian-during-smooth-pursuit/</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:crowding_at_end_pooling.png&diff=36050File:crowding at end pooling.png2018-04-03T04:40:01Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=End-to-End_Differentiable_Adversarial_Imitation_Learning&diff=36049End-to-End Differentiable Adversarial Imitation Learning2018-04-03T04:25:35Z<p>J3ku: Clarify results in experiment table (where is the boldface?)</p>
<hr />
<div>= Introduction =<br />
The ability to imitate an expert policy is very beneficial in the case of automating human demonstrated tasks. Assuming that a sequence of state action pairs (trajectories) of an expert policy are available, a new policy can be trained that imitates the expert without having access to the original reward signal used by the expert. There are two main approaches to solve the problem of imitating a policy; they are Behavioural Cloning (BC) and Inverse Reinforcement Learning (IRL). BC directly learns the conditional distribution of actions over states in a supervised fashion by training on single time-step state-action pairs. The disadvantage of BC is that the training requires large amounts of expert data, which is hard to obtain. In addition, an agent trained using BC is unaware of how its action can affect future state distribution. The second method using IRL involves recovering a reward signal under which the expert is uniquely optimal; the main disadvantage is that it’s an ill-posed problem.<br />
<br />
To address the problem of imitating an expert policy, techniques based on Generative Adversarial Networks (GANs) have been proposed in recent years. GANs use a discriminator to guide the generative model towards producing patterns like those of the expert. The generator is guided as it tries to produce samples on the correct side of the discriminators decision boundary hyper-plane, as seen in Figure 1. This idea was used by (Ho & Ermon, 2016) in their work titled Generative Adversarial Imitation Learning (GAIL) to imitate an expert policy in a model-free setup. A model free setup is the one where the agent cannot make predictions about what the next state and reward will be before it takes each action since the transition function to move from state A to state B is not learned. <br />
<br />
The disadvantage of the model-free approach comes to light when training stochastic policies. The presence of stochastic elements breaks the flow of information (gradients) from one neural network to the other, thus prohibiting the use of backpropagation. In this situation, a standard solution is to use gradient estimation (Williams, 1992). This tends to suffer from high variance, resulting in a need for larger sample sizes as well as variance reduction methods. This paper proposes a model-based imitation learning algorithm (MGAIL), in which information propagates from the guiding neural network (D) to the generative model (G), which in this case represents the policy <math>\pi</math> that is to be trained. Training policy <math>\pi</math> assumes the existence of an expert policy <math>\pi_{E}</math> with given trajectories <math>\{s_{0},a_{0},s_{1},...\}^{N}_{i=0}</math> which it aims to imitate without access to the original reward signal <math>r_{e}</math>. This is achieved by two steps: (1) learning a forward model that approximates the environment’s dynamics (2) building an end-to-end differentiable computation graph that spans over multiple time-steps. The gradient in such a graph carries information from future states to earlier time-steps, helping the policy to account for compounding errors.<br />
<br />
<br />
[[File:GeneratorFollowingDiscriminator.png|center]]<br />
<br />
Figure 1: '''Illustration of GANs.''' The generative model follows the discriminating hyper-plane defined by the discriminator. Eventually, G will produce patterns similar to the expert patterns.<br />
<br />
= Background =<br />
== Markov Decision Process ==<br />
Consider an infinite-horizon discounted Markov decision process (MDP), defined by the tuple <math>(S, A, P, r, \rho_0, \gamma)</math> where <math>S</math> is the set of states, <math>A</math> is a set of actions, <math>P :<br />
S × A × S → [0, 1]</math> is the transition probability distribution, <math>r : (S × A) → R</math> is the reward function, <math>\rho_0 : S → [0, 1]</math> is the distribution over initial states, and <math>γ ∈ (0, 1)</math> is the discount factor. Let <math>π</math> denote a stochastic policy <math>π : S × A → [0, 1]</math>, <math>R(π)</math> denote its expected discounted reward: <math>E_πR = E_π [\sum_{t=0}^T \gamma^t r_t]</math> and <math>τ</math> denote a trajectory of states and actions <math>τ = {s_0, a_0, s_1, a_1, ...}</math>.<br />
<br />
== Imitation Learning ==<br />
A common technique for performing imitation learning is to train a policy <math> \pi </math> that minimizes some loss function <math> l(s, \pi(s)) </math> with respect to a discounted state distribution encountered by the expert: <math> d_\pi(s) = (1-\gamma)\sum_{t=0}^{\infty}\gamma^t p(s_t) </math>. This can be obtained using any supervised learning (SL) algorithm, but the policy's prediction affects future state distributions; this violates the independent and identically distributed (i.i.d) assumption made by most SL algorithms. This process is susceptible to compounding errors since a slight deviation in the learner's behavior can lead to different state distributions not encountered by the expert policy. <br />
<br />
This issue was overcome through the use of the Forward Training (FT) algorithm which trains a non-stationary policy iteratively over time. At each time step a new policy is trained on the state distribution induced by the previously trained policies <math>\pi_0</math>, <math>\pi_1</math>, ...<math>\pi_{t-1}</math>. This is continued till the end of the time horizon to obtain a policy that can mimic the expert policy. This requirement to train a policy at each time step till the end makes the FT algorithm impractical for cases where the time horizon is very large or undefined. This shortcoming is resolved using the Stochastic Mixing Iterative Learning (SMILe) algorithm. SMILe trains a stochastic stationary policy over several iterations under the trajectory distribution induced by the previously trained policy: <math> \pi_t = \pi_{t-1} + \alpha (1 - \alpha)^{t-1}(\hat{\pi}_t - \pi_0)</math>, with <math>\pi_0</math> following expert's policy at the start of training.<br />
<br />
== Generative Adversarial Networks ==<br />
GANs learn a generative model that can fool the discriminator by using a two-player zero-sum game:<br />
<br />
\begin{align} <br />
\underset{G}{\operatorname{argmin}}\; \underset{D\in (0,1)}{\operatorname{argmax}} = \mathbb{E}_{x\sim p_E}[log(D(x)]\ +\ \mathbb{E}_{z\sim p_z}[log(1 - D(G(z)))]<br />
\end{align}<br />
<br />
In the above equation, <math> p_E </math> represents the expert distribution and <math> p_z </math> represents the input noise distribution from which the input to the generator is sampled. The generator produces patterns and the discriminator judges if the pattern was generated or from the expert data. When the discriminator cannot distinguish between the two distributions the game ends and the generator has learned to mimic the expert. GANs rely on basic ideas such as binary classification and algorithms such as backpropagation in order to learn the expert distribution.<br />
<br />
GAIL applies GANs to the task of imitating an expert policy in a model-free approach. GAIL uses similar objective functions like GANs, but the expert distribution in GAIL represents the joint distribution over state action tuples:<br />
<br />
\begin{align} <br />
\underset{\pi}{\operatorname{argmin}}\; \underset{D\in (0,1)}{\operatorname{argmax}} = \mathbb{E}_{\pi}[log(D(s,a)]\ +\ \mathbb{E}_{\pi_E}[log(1 - D(s,a))] - \lambda H(\pi))<br />
\end{align}<br />
<br />
where <math> H(\pi) \triangleq \mathbb{E}_{\pi}[-log\: \pi(a|s)]</math> is the entropy.<br />
<br />
This problem cannot be solved using the standard methods described for GANs because the generator in GAIL represents a stochastic policy. The exact form of the first term in the above equation is given by: <math> \mathbb{E}_{s\sim \rho_\pi(s)}\mathbb{E}_{a\sim \pi(\cdot |s)} [log(D(s,a)] </math>.<br />
<br />
The two-player game now depends on the stochastic properties (<math> \theta </math>) of the policy, and it is unclear how to differentiate the above equation with respect to <math> \theta </math>. This problem can be overcome using score functions such as REINFORCE to obtain an unbiased gradient estimation:<br />
<br />
\begin{align}<br />
\nabla_\theta\mathbb{E}_{\pi} [log\; D(s,a)] \cong \hat{\mathbb{E}}_{\tau_i}[\nabla_\theta\; log\; \pi_\theta(a|s)Q(s,a)]<br />
\end{align}<br />
<br />
where <math> Q(\hat{s},\hat{a}) </math> is the score function of the gradient:<br />
<br />
\begin{align}<br />
Q(\hat{s},\hat{a}) = \hat{\mathbb{E}}_{\tau_i}[log\; D(s,a) | s_0 = \hat{s}, a_0 = \hat{a}]<br />
\end{align}<br />
<br />
<br />
REINFORCE gradients suffer from high variance which makes them difficult to work with even after applying variance reduction techniques. While recent general variance reduction techniques like RELAX (Grathwohl et al., 2017) work well, they rely on multiple evaluations of the loss function or learning a surrogate neural network. Unfortunately, this is too computationally difficult for our task. In order to better understand the changes required to fool the discriminator we need access to the gradients of the discriminator network, which can be obtained from the Jacobian of the discriminator. This paper demonstrates the use of a forward model along with the Jacobian of the discriminator to train a policy, without using high-variance gradient estimations.<br />
<br />
= Algorithm =<br />
This section first analyzes the characteristics of the discriminator network, then describes how a forward model can enable policy imitation through GANs. Lastly, the model based adversarial imitation learning algorithm is presented.<br />
<br />
== The discriminator network ==<br />
The discriminator network is trained to predict the conditional distribution: <math> D(s,a) = p(y|s,a) </math> where <math> y \in (\pi_E, \pi) </math>.<br />
<br />
The discriminator is trained on an even distribution of expert and generated examples; hence <math> p(\pi) = p(\pi_E) = \frac{1}{2} </math>. Given this and applying Bayes' theorem, we can rearrange and factor <math> D(s,a) </math> to obtain:<br />
<br />
\begin{aligned}<br />
D(s,a) &= p(\pi|s,a) \\<br />
& = \frac{p(s,a|\pi)p(\pi)}{p(s,a|\pi)p(\pi) + p(s,a|\pi_E)p(\pi_E)} \\<br />
& = \frac{p(s,a|\pi)}{p(s,a|\pi) + p(s,a|\pi_E)} \\<br />
& = \frac{1}{1 + \frac{p(s,a|\pi_E)}{p(s,a|\pi)}} \\<br />
& = \frac{1}{1 + \frac{p(a|s,\pi_E)}{p(a|s,\pi)} \cdot \frac{p(s|\pi_E)}{p(s|\pi)}} \\<br />
\end{aligned}<br />
<br />
Define <math> \varphi(s,a) </math> and <math> \psi(s) </math> to be:<br />
<br />
\begin{aligned}<br />
\varphi(s,a) = \frac{p(a|s,\pi_E)}{p(a|s,\pi)}, \psi(s) = \frac{p(s|\pi_E)}{p(s|\pi)}<br />
\end{aligned}<br />
<br />
to get the final expression for <math> D(s,a) </math>:<br />
\begin{aligned}<br />
D(s,a) = \frac{1}{1 + \varphi(s,a)\cdot \psi(s)}<br />
\end{aligned}<br />
<br />
<math> \varphi(s,a) </math> represents a policy likelihood ratio, and <math> \psi(s) </math> represents a state distribution likelihood ratio. Based on these expressions, the paper states that the discriminator makes its decisions by answering two questions. The first question relates to state distribution: what is the likelihood of encountering state <math> s </math> under the distribution induces by <math> \pi_E </math> vs <math> \pi </math>? The second question is about behavior: given a state <math> s </math>, how likely is action a under <math> \pi_E </math> vs <math> \pi </math>? The desired change in state is given by <math> \psi_s \equiv \partial \psi / \partial s </math>; this information can by obtained from the partial derivatives of <math> D(s,a) </math>, which is why these derivatives are proposed to be used for training policies (see following sections):<br />
<br />
\begin{aligned}<br />
\nabla_aD &= - \frac{\varphi_a(s,a)\psi(s)}{(1 + \varphi(s,a)\psi(s))^2} \\<br />
\nabla_sD &= - \frac{\varphi_s(s,a)\psi(s) + \varphi(s,a)\psi_s(s)}{(1 + \varphi(s,a)\psi(s))^2} \\<br />
\end{aligned}<br />
<br />
== Backpropagating through stochastic units ==<br />
There is interest in training stochastic policies because stochasticity encourages exploration for Policy Gradient methods. This is a problem for algorithms that build differentiable computation graphs where the gradients flow from one component to another since it is unclear how to backpropagate through stochastic units. The following subsections show how to estimate the gradients of continuous and categorical stochastic elements for continuous and discrete action domains respectively.<br />
<br />
=== Continuous Action Distributions ===<br />
In the case of continuous action policies, re-parameterization was used to enable computing the derivatives of stochastic models. Assuming that the stochastic policy has a Gaussian distribution <math> \mathcal{N}(\mu_{\theta} (s), \sigma_{\theta}^2 (s))</math>, where the mean and variance are given by some deterministic functions <math>\mu_{\theta}</math> and <math>\sigma_{\theta}</math>, then the policy <math> \pi </math> can be written as <math> \pi_\theta(a|s) = \mu_\theta(s) + \xi \sigma_\theta(s) </math>, where <math> \xi \sim N(0,1) </math>. This way, the authors are able to get a Monte-Carlo estimator of the derivative of the expected value of <math> D(s, a) </math> with respect to <math> \theta </math>:<br />
<br />
\begin{align}<br />
\nabla_\theta\mathbb{E}_{\pi(a|s)}D(s,a) = \mathbb{E}_{\rho (\xi )}\nabla_a D(a,s) \nabla_\theta \pi_\theta(a|s) \cong \frac{1}{M}\sum_{i=1}^{M} \nabla_a D(s,a) \nabla_\theta \pi_\theta(a|s)\Bigr|_{\substack{\xi=\xi_i}}<br />
\end{align}<br />
<br />
=== Categorical Action Distributions ===<br />
In the case of discrete action domains, the paper uses categorical re-parameterization with Gumbel-Softmax. This method relies on the Gumbel-Max trick which is a method for drawing samples from a categorical distribution with class probabilities <math> \pi(a_1|s),\pi(a_2|s),...,\pi(a_N|s) </math>:<br />
<br />
\begin{align}<br />
a_{argmax} = \underset{i}{argmax}[g_i + log\ \pi(a_i|s)]\textrm{, where } g_i \sim Gumbel(0, 1).<br />
\end{align}<br />
<br />
Gumbel-Softmax provides a differentiable approximation of the samples obtained using the Gumbel-Max trick (Gumbel-softmax allows us to generate a differentiable sample from a discrete distribution, which is needed in this trajectory imitation setting.):<br />
<br />
\begin{align}<br />
a_{softmax} = \frac{exp[\frac{1}{\tau}(g_i + log\ \pi(a_i|s))]}{\sum_{j=1}^{k}exp[\frac{1}{\tau}(g_j + log\ \pi(a_i|s))]}<br />
\end{align}<br />
<br />
<br />
In the above equation, the hyper-parameter <math> \tau </math> (temperature) trades bias for variance. When <math> \tau </math> gets closer to zero, the softmax operator acts like argmax resulting in a low bias, but high variance; vice versa when the <math> \tau </math> is large.<br />
<br />
The authors use <math> a_{softmax} </math> to interact with the environment; argmax is applied over <math> a_{softmax} </math> to obtain a single “pure” action, but the continuous approximation is used in the backward pass using the estimation: <math> \nabla_\theta\; a_{argmax} \approx \nabla_\theta\; a_{softmax} </math>.<br />
<br />
== Backpropagating through a Forward model ==<br />
The above subsections presented the means for extracting the partial derivative <math> \nabla_aD </math>. The main contribution of this paper is incorporating the use of <math> \nabla_sD </math>. In a model-free approach the state <math> s </math> is treated as a fixed input, therefore <math> \nabla_sD </math> is discarded. This is illustrated in Figure 2. This work uses a model-based approach which makes incorporating <math> \nabla_sD </math> more involved. In the model-based approach, a state <math> s_t </math> can be written as a function of the previous state action pair: <math> s_t = f(s_{t-1}, a_{t-1}) </math>, where <math> f </math> represents the forward model. Using the forward model and the law of total derivatives we get:<br />
<br />
\begin{align}<br />
\nabla_\theta D(s_t,a_t)\Bigr|_{\substack{s=s_t, a=a_t}} &= \frac{\partial D}{\partial a}\frac{\partial a}{\partial \theta}\Bigr|_{\substack{a=a_t}} + \frac{\partial D}{\partial s}\frac{\partial s}{\partial \theta}\Bigr|_{\substack{s=s_t}} \\<br />
&= \frac{\partial D}{\partial a}\frac{\partial a}{\partial \theta}\Bigr|_{\substack{a=a_t}} + \frac{\partial D}{\partial s}\left (\frac{\partial f}{\partial s}\frac{\partial s}{\partial \theta}\Bigr|_{\substack{s=s_{t-1}}} + \frac{\partial f}{\partial a}\frac{\partial a}{\partial \theta}\Bigr|_{\substack{a=a_{t-1}}} \right )<br />
\end{align}<br />
<br />
<br />
Using this formula, the error regarding deviations of future states <math> (\psi_s) </math> propagate back in time and influence the actions of policies in earlier times. This is summarized in Figure 3.<br />
<br />
[[File:modelFree_blockDiagram.PNG|400px|center]]<br />
<br />
Figure 2: Block-diagram of the model-free approach: given a state <math> s </math>, the policy outputs <math> \mu </math> which is fed to a stochastic sampling unit. An action <math> a </math> is sampled, and together with <math> s </math> are presented to the discriminator network. In the backward phase, the error message <math> \delta_a </math> is blocked at the stochastic sampling unit. From there, a high-variance gradient estimation is used (<math> \delta_{HV} </math>). Meanwhile, the error message <math> \delta_s </math> is flushed.<br />
<br />
[[File:modelBased_blockDiagram.PNG|700px|center]]<br />
<br />
Figure 3: Block diagram of model-based adversarial imitation learning. <br />
<br />
Figure 3 describes the computation graph for training the policy (i.e. G). The discriminator network D is fixed at this stage and is trained separately. At time <math> t </math> of the forward pass, <math> \pi </math> outputs a distribution over actions: <math> \mu_t = \pi(s_t) </math>, from which an action at is sampled. For example, in the continuous case, this is done using the re-parametrization trick: <math> a_t = \mu_t + \xi \cdot \sigma </math>, where <math> \xi \sim N(0,1) </math>. The next state <math> s_{t+1} = f(s_t, a_t) </math> is computed using the forward model (which is also trained separately), and the entire process repeats for time <math> t+1 </math>. In the backward pass, the gradient of <math> \pi </math> is comprised of a.) the error message <math> \delta_a </math> (Green) that propagates fluently through the differentiable approximation of the sampling process. And b.) the error message <math> \delta_s </math> (Blue) of future time-steps, that propagate back through the differentiable forward model.<br />
<br />
== MGAIL Algorithm ==<br />
Shalev- Shwartz et al. (2016) and Heess et al. (2015) built a multi-step computation graph for describing the familiar policy gradient objective; in this case it is given by:<br />
<br />
\begin{align}<br />
J(\theta) = \mathbb{E}\left [ \sum_{t=0}^{T} \gamma ^t D(s_t,a_t)|\theta\right ]<br />
\end{align}<br />
<br />
<br />
Using the results from Heess et al. (2015) this paper demonstrates how to differentiate <math> J(\theta) </math> over a trajectory of <math>(s,a,s’) </math> transitions:<br />
<br />
\begin{align}<br />
J_s &= \mathbb{E}_{p(a|s)}\mathbb{E}_{p(s'|s,a)}\left [ D_s + D_a \pi_s + \gamma J'_{s'}(f_s + f_a \pi_s) \right] \\<br />
J_\theta &= \mathbb{E}_{p(a|s)}\mathbb{E}_{p(s'|s,a)}\left [ D_a \pi_\theta + \gamma (J'_{s'} f_a \pi_\theta + J'_\theta) \right]<br />
\end{align}<br />
<br />
The policy gradient <math> \nabla_\theta J </math> is calculated by applying equations 12 and 13 recursively for <math> T </math> iterations. The MGAIL algorithm is presented below.<br />
<br />
[[File:MGAIL_alg.PNG]]<br />
<br />
== Forward Model Structure ==<br />
The stability of the learning process depends on the prediction accuracy of the forward model, but learning an accurate forward model is challenging by itself. The authors propose methods for improving the performance of the forward model based on two aspects of its functionality. First, the forward model should learn to use the action as an operator over the state space. To accomplish this, the actions and states, which are sampled form different distributions need to be first represented in a shared space. This is done by encoding the state and action using two separate neural networks and combining their outputs to form a single vector. Additionally, multiple previous states are used to predict the next state by representing the environment as an <math> n^{th} </math> order MDP. A gated recurrent units (GRU, a simpler variant on the LSTM model) layer is incorporated into the state encoder to enable recurrent connections from previous states. Using these modifications, the model is able to achieve better, and more stable results compared to the standard forward model based on a feed forward neural network. The comparison is presented in Figure 4.<br />
<br />
[[File:performance_comparison.PNG]]<br />
<br />
Figure 4: Performance comparison between a basic forward model (Blue), and the advanced forward model (Green).<br />
<br />
= Experiments =<br />
The proposed algorithm is evaluated on three discrete control tasks (Cartpole, Mountain-Car, Acrobot) and five continuous control tasks (Hopper, Walker, Half-Cheetah, Ant, and Humanoid). These tasks are modelled by the MuJoCo physics simulator (Todorov et al., 2012), contain second order dynamics and utilize direct torque control. Expert policies are trained using the Trust Region Policy Optimization (TRPO) algorithm (Schulman et al., 2015). Different number of trajectories are used to train the expert for each task, but all trajectories are of length 1000.<br />
The discriminator and generator (policy) networks contains two hidden layers with ReLU non-linearities and are trained using the ADAM optimizer. The total reward received over a period of <math> N </math> steps using BC, GAIL and MGAIL is presented in Table 1. The proposed algorithm achieved the highest reward for most environments while exhibiting performance comparable to the expert over all of them. A comparison between the basic forward model and the more advanced forward model is also made and described in the previous section of this summary. The two models compared are shown below.<br />
<br />
[[File:baram17_forward.PNG]]<br />
<br />
[[File:mgail_test_results_1.PNG]]<br />
<br />
[[File:mgail_test_results.PNG]]<br />
<br />
Table 1. Policy performance, <math> \pm </math> represents one standard deviation, a higher (reward) value is better. MGAIL consistently outperforms both GAIL and Behavioural cloning approaches, except on the Cartpole, where MGAIL and GAIL perform equally.<br />
<br />
= Discussion =<br />
This paper presented a model-free algorithm for imitation learning. It demonstrated how a forward model can be used to train policies using the exact gradient of the discriminator network. A downside of this approach is the need to learn a forward model; this could be difficult in certain domains. Learning the system dynamics directly from raw images is considered as one line of future work. Another future work is to address the violation of the fundamental assumption made by all supervised learning algorithms, which requires the data to be i.i.d. This problem arises because the discriminator and forward models are trained in a supervised learning fashion using data sampled from a dynamic distribution. The authors tried a solution proposed by another paper (Loshchilov & Hutter, 2016), which is to reset the learning rate several times during training period, but it did not result in significant improvements.<br />
<br />
= Implementation =<br />
The following repository provides the source code for the paper: https://github.com/itaicaspi/mgail. The repository provides the source code written by the authors, in Tensorflow.<br />
<br />
= Source =<br />
# Baram, Nir, et al. "End-to-end differentiable adversarial imitation learning." International Conference on Machine Learning. 2017.<br />
# Ho, Jonathan, and Stefano Ermon. "Generative adversarial imitation learning." Advances in Neural Information Processing Systems. 2016.<br />
# Shalev-Shwartz, Shai, et al. "Long-term planning by short-term prediction." arXiv preprint arXiv:1602.01580 (2016).<br />
# Heess, Nicolas, et al. "Learning continuous control policies by stochastic value gradients." Advances in Neural Information Processing Systems. 2015.<br />
# Schulman, John, et al. "Trust region policy optimization." International Conference on Machine Learning. 2015.<br />
# Caspi, I. (n.d.). Itaicaspi/mgail. Retrieved March 25, 2018, from https://github.com/itaicaspi/mgail.<br />
# Grathwohl, W., Choi, D., Wu, Y., Roeder, G., & Duvenaud, D. (2017). Backpropagation through the Void: Optimizing control variates for black-box gradient estimation. arXiv preprint arXiv:1711.00123.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/IMPROVING_GANS_USING_OPTIMAL_TRANSPORT&diff=36048stat946w18/IMPROVING GANS USING OPTIMAL TRANSPORT2018-04-03T04:15:33Z<p>J3ku: Add to CIFAR-10 experiment</p>
<hr />
<div>== Introduction ==<br />
Recently, the problem of how to learn models that generate media such as images, video, audio and text has become very popular and is called Generative Modeling. One of the main benefits of such an approach is that generative models can be trained on unlabeled data that is readily available . Therefore, generative networks have a huge potential in the field of deep learning.<br />
<br />
Generative Adversarial Networks (GANs) are powerful generative models used for unsupervised learning techniques where the 2 agents compete to generate a zero-sum model. A GAN model consists of a generator and a discriminator or critic. The generator is a neural network which is trained to generate data having a distribution matched with the distribution of the real data. The critic is also a neural network, which is trained to separate the generated data from the real data. A loss function that measures the distribution distance between the generated data and the real one is important to train the generator.<br />
<br />
Optimal transport theory, which is another approach to measuring distances between distributions, evaluates the distribution distance between the generated data and the training data based on a metric, which provides another method for generator training. The main advantage of optimal transport theory over the distance measurement in GAN is its closed form solution for having a tractable training process. But the theory might also result in inconsistency in statistical estimation due to the given biased gradients if the mini-batches method is applied (Bellemare et al.,<br />
2017).<br />
<br />
This paper presents a variant GANs named OT-GAN, which incorporates a discriminative metric called 'Mini-batch Energy Distance' into its critic in order to overcome the issue of biased gradients.<br />
<br />
== GANs and Optimal Transport ==<br />
<br />
===Generative Adversarial Nets===<br />
Original GAN was firstly reviewed. The objective function of the GAN: <br />
<br />
[[File:equation1.png|700px]]<br />
<br />
The goal of GANs is to train the generator g and the discriminator d finding a pair of (g,d) to achieve Nash equilibrium(such that either of them cannot reduce their cost without changing the others' parameters). However, it could cause failure of converging since the generator and the discriminator are trained based on gradient descent techniques.<br />
<br />
===Wasserstein Distance (Earth-Mover Distance)===<br />
<br />
In order to solve the problem of convergence failure, Arjovsky et. al. (2017) suggested Wasserstein distance (Earth-Mover distance) based on the optimal transport theory.<br />
<br />
[[File:equation2.png|600px]]<br />
<br />
where <math> \prod (p,g) </math> is the set of all joint distributions <math> \gamma (x,y) </math> with marginals <math> p(x) </math> (real data), <math> g(y) </math> (generated data). <math> c(x,y) </math> is a cost function and the Euclidean distance was used by Arjovsky et. al. in the paper. <br />
<br />
The Wasserstein distance can be considered as moving the minimum amount of points between distribution <math> g(y) </math> and <math> p(x) </math> such that the generator distribution <math> g(y) </math> is similar to the real data distribution <math> p(x) </math>.<br />
<br />
Computing the Wasserstein distance is intractable. The proposed Wasserstein GAN (W-GAN) provides an estimated solution by switching the optimal transport problem into Kantorovich-Rubinstein dual formulation using a set of 1-Lipschitz functions. A neural network can then be used to obtain an estimation.<br />
<br />
[[File:equation3.png|600px]]<br />
<br />
W-GAN helps to solve the unstable training process of original GAN and it can solve the optimal transport problem approximately, but it is still intractable.<br />
<br />
===Sinkhorn Distance===<br />
Genevay et al. (2017) proposed to use the primal formulation of optimal transport instead of the dual formulation to generative modeling. They introduced Sinkhorn distance which is a smoothed generalization of the Wasserstein distance.<br />
[[File: equation4.png|600px]]<br />
<br />
It introduced entropy restriction (<math> \beta </math>) to the joint distribution <math> \prod_{\beta} (p,g) </math>. This distance could be generalized to approximate the mini-batches of data <math> X ,Y</math> with <math> K </math> vectors of <math> x, y</math>. The <math> i, j </math> th entry of the cost matrix <math> C </math> can be interpreted as the cost it needs to transport the <math> x_i </math> in mini-batch X to the <math> y_i </math> in mini-batch <math>Y </math>. The resulting distance will be:<br />
<br />
[[File: equation5.png|550px]]<br />
<br />
where <math> M </math> is a <math> K \times K </math> matrix, each row of <math> M </math> is a joint distribution of <math> \gamma (x,y) </math> with positive entries. The summmation of rows or columns of <math> M </math> is always equal to 1. <br />
<br />
This mini-batch Sinkhorn distance is not only fully tractable but also capable of solving the instability problem of GANs. However, it is not a valid metric over probability distribution when taking the expectation of <math> \mathcal{W}_{c} </math> and the gradients are biased when the mini-batch size is fixed.<br />
<br />
===Energy Distance (Cramer Distance)===<br />
In order to solve the above problem, Bellemare et al. proposed Energy distance:<br />
<br />
[[File: equation6.png|700px]]<br />
<br />
where <math> x, x' </math> and <math> y, y'</math> are independent samples from data distribution <math> p </math> and generator distribution <math> g </math>, respectively. Based on the Energy distance, Cramer GAN is to minimize the ED distance metric when training the generator.<br />
<br />
==Mini-Batch Energy Distance==<br />
Salimans et al. (2016) mentioned that comparing to use distributions over individual images, mini-batch GAN is more powerful when using the distributions over mini-batches <math> g(X), p(X) </math>. The distance measure is generated for mini-batches.<br />
<br />
===Generalized Energy Distance===<br />
The generalized energy distance allowed to use non-Euclidean distance functions d. It is also valid for mini-batches and is considered better than working with individual data batch.<br />
<br />
[[File: equation7.png|670px]]<br />
<br />
Similarly as defined in the Energy distance, <math> X, X' </math> and <math> Y, Y'</math> can be the independent samples from data distribution <math> p </math> and the generator distribution <math> g </math>, respectively. While in Generalized engergy distance, <math> X, X' </math> and <math> Y, Y'</math> can also be valid for mini-batches. The <math> D_{GED}(p,g) </math> is a metric when having <math> d </math> as a metric. Thus, taking the triangle inequality of <math> d </math> into account, <math> D(p,g) \geq 0,</math> and <math> D(p,g)=0 </math> when <math> p=g </math>.<br />
<br />
===Mini-Batch Energy Distance===<br />
As <math> d </math> is free to choose, authors proposed Mini-batch Energy Distance by using entropy-regularized Wasserstein distance as <math> d </math>. <br />
<br />
[[File: equation8.png|650px]]<br />
<br />
where <math> X, X' </math> and <math> Y, Y'</math> are independent sampled mini-batches from the data distribution <math> p </math> and the generator distribution <math> g </math>, respectively. This distance metric combines the energy distance with primal form of optimal transport over mini-batch distributions <math> g(Y) </math> and <math> p(X) </math>. Inside the generalized energy distance, the Sinkhorn distance is a valid metric between each mini-batches. By adding the <math> - \mathcal{W}_c (Y,Y')</math> and <math> \mathcal{W}_c (X,Y)</math> to equation (5) and using energy distance, the objective becomes statistically consistent (meaning the objective converges to the true parameter value for large sample sizes) and mini-batch gradients are unbiased.<br />
<br />
==Optimal Transport GAN (OT-GAN)==<br />
<br />
The mini-batch energy distance which was proposed depends on the transport cost function <math>c(x,y)</math>. One possibility would be to choose c to be some fixed function over vectors, like Euclidean distance, but the authors found this to perform poorly in preliminary experiments. For simple fixed cost functions like Euclidean distance, there exists many bad distributions <math>g</math> in higher dimensions for which the mini-batch energy distance is zero such that it is difficult to tell <math>p</math> and <math>g</math> apart if the sample size is not big enough. To solve this the authors propose learning the cost function adversarially, so that it can adapt to the generator distribution <math>g</math> and thereby become more discriminative. <br />
<br />
In practice, in order to secure the statistical efficiency (i.e. being able to tell <math>p</math> and <math>g</math> apart without requiring an enormous sample size when their distance is close to zero), authors suggested using cosine distance between vectors <math> v_\eta (x) </math> and <math> v_\eta (y) </math> based on the deep neural network that maps the mini-batch data to a learned latent space. Here is the transportation cost:<br />
<br />
[[File: euqation9.png|370px]]<br />
<br />
where the <math> v_\eta </math> is chosen to maximize the resulting minibatch energy distance.<br />
<br />
Unlike the practice when using the original GANs, the generator was trained more often than the critic, which keep the cost function from degeneration. The resulting generator in OT-GAN has a well defined and statistically consistent objective through the training process.<br />
<br />
The algorithm is defined below. The backpropagation is not used in the algorithm since ignoring this gradient flow is justified by the envelope theorem (i.e. when changing the parameters of the objective function, changes in the optimizer do not contribute to a change in the objective function). Stochastic gradient descent is used as the optimization method in algorithm 1 below, although other optimizers are also possible. In fact, Adam was used in experiments. <br />
<br />
[[File: al.png|600px]]<br />
<br />
<br />
[[File: al_figure.png|600px]]<br />
<br />
==Experiments==<br />
<br />
In order to demonstrate the supermum performance of the OT-GAN, authors compared it with the original GAN and other popular models based on four experiments: Dataset recovery; CIFAR-10 test; ImageNet test; and the conditional image synthesis test.<br />
<br />
===Mixture of Gaussian Dataset===<br />
OT-GAN has a statistically consistent objective when it is compared with the original GAN (DC-GAN), such that the generator would not update to a wrong direction even if the signal provided by the cost function to the generator is not good. In order to prove this advantage, authors compared the OT-GAN with the original GAN loss (DAN-S) based on a simple task. The task was set to recover all of the 8 modes from 8 Gaussian mixers in which the means were arranged in a circle. MLP with RLU activation functions were used in this task. The critic was only updated for 15K iterations. The generator distribution was tracked for another 25K iteration. The results showed that the original GAN experiences the model collapse after fixing the discriminator while the OT-GAN recovered all the 8 modes from the mixed Gaussian data.<br />
<br />
[[File: 5_1.png|600px]]<br />
<br />
===CIFAR-10===<br />
<br />
The dataset CIFAR-10 was then used for inspecting the effect of batch-size to the model training process and the image quality. OT-GAN and four other methods were compared using "inception score" as the criteria for comparison. Figure 3 shows the change of inceptions scores (y-axis) by the increased of the iteration number. Scores of four different batch sizes (200, 800, 3200 and 8000) were compared. The results show that a larger batch size, which would more likely cover more modes in the distribution of data, lead to a more stable model showing a larger value in inception score. However, a large batch size would also require a high-performance computational environment. The sample quality across all 5 methods, ran using a batch size of 8000, are compared in Table 1 where the OT_GAN has the best score.<br />
<br />
The OT-GAN was trained using Adam optimizer. The learning rate was set to <math> 3 x10^-4, β_1 = 0.5, β_2 = 0.999 </math>. The introduced OT-GAN algorithm also includes two additional hyperparameters for the Sinkhorn algorithm. The first hyperparameters indicated the number of iterations to run the algorithm and the second <math> 1/λ </math> the entropy penalty of alignments. The authors found that a value of 500 worked well for both mentioned hyperparameters.<br />
<br />
[[File: 5_2.png|600px]]<br />
<br />
Figure 4 below shows samples generated by the OT-GAN trained with a batch size of 8000. Figure 5 below shows random samples from a model trained with the same architecture and hyperparameters, but with random matching of samples in place of optimal transport.<br />
<br />
[[File: ot_gan_cifar_10_samples.png|600px]]<br />
<br />
===ImageNet Dogs===<br />
<br />
In order to investigate the performance of OT-GAN when dealing with the high-quality images, the dog subset of ImageNet (128*128) was used to train the model. Figure 6 shows that OT-GAN produces less nonsensical images and it has a higher inception score compare to the DC-GAN. <br />
<br />
[[File: 5_3.png|600px]]<br />
<br />
<br />
To analyze mode collapse in GANs the authors trained both types of GANs for a large number of epochs. They find the DCGAN shows mode collapse as soon as 900 epochs. They trained the OT-GAN for 13000 epochs and saw no evidence of mode collapse or less diversity in the samples. Samples can be viewed in Figure 9.<br />
<br />
[[File: ModelCollapseImageNetDogs.png|600px]]<br />
<br />
===Conditional Generation of Birds===<br />
<br />
The last experiment was to compare OT-GAN with three popular GAN models for processing the text-to-image generation demonstrating the performance on conditional image synthesis. As can be found from Table 2, OT-GAN received the highest inception score than the scores of the other three models. <br />
<br />
[[File: 5_4.png|600px]]<br />
<br />
The algorithm used to obtain the results above is conditional generation generalized from '''Algorithm 1''' to include conditional information <math>s</math> such as some text description of an image. The modified algorithm is outlined in '''Algorithm 2'''.<br />
<br />
[[File: paper23_alg2.png|600px]]<br />
<br />
==Conclusion==<br />
<br />
In this paper, an OT-GAN method was proposed based on the optimal transport theory. A distance metric that combines the primal form of the optimal transport and the energy distance was given was presented for realizing the OT-GAN. The results showed OT-GAN to be uniquely stable when trained with large mini batches and state of the art results were achieved on some datasets. One of the advantages of OT-GAN over other GAN models is that OT-GAN can stay on the correct track with an unbiased gradient even if the training on critic is stopped or presents a weak cost signal. The performance of the OT-GAN can be maintained when the batch size is increasing, though the computational cost has to be taken into consideration.<br />
<br />
==Critique==<br />
<br />
The paper presents a variant of GANs by defining a new distance metric based on the primal form of optimal transport and the mini-batch energy distance. The stability was demonstrated through the four experiments that comparing OP-GAN with other popular methods. However, limitations in computational efficiency were not discussed much. Furthermore, in section 2, the paper lacks explanation on using mini-batches instead of a vector as input when applying Sinkhorn distance. It is also confusing when explaining the algorithm in section 4 about choosing M for minimizing <math> \mathcal{W}_c </math>. Lastly, it is found that it is lack of parallel comparison with existing GAN variants in this paper. Readers may feel jumping from one algorithm to another without necessary explanations.<br />
<br />
==Reference==<br />
Salimans, Tim, Han Zhang, Alec Radford, and Dimitris Metaxas. "Improving GANs using optimal transport." (2018).</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:ot_gan_cifar_10_samples.png&diff=36047File:ot gan cifar 10 samples.png2018-04-03T04:11:08Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/MaskRNN:_Instance_Level_Video_Object_Segmentation&diff=36046stat946w18/MaskRNN: Instance Level Video Object Segmentation2018-04-03T04:05:49Z<p>J3ku: Add major challenges, add reason for online fine-tuning</p>
<hr />
<div>== Introduction ==<br />
Deep Learning has produced state of the art results in many computer vision tasks like image classification, object localization, object detection, object segmentation, semantic segmentation and instance level video object segmentation. Image classification classify the image based on the prominent objects. Object localization is the task of finding objects’ location in the frame. Object Segmentation task involves providing a pixel map which represents the pixel wise location of the objects in the image. Semantic segmentation task attempts at segmenting the image into meaningful parts. Instance level video object segmentation is the task of consistent object segmentation in video sequences. Deforming shapes, fast movements, and occlusion from multiple objects, are just some of the significant challenges in instance level video object segmentation.<br />
<br />
There are 2 different types of video object segmentation: Unsupervised and Semi-supervised. In unsupervised video object segmentation, the task is to find the salient objects and track the main objects in the video. In an unsupervised setting, the ground truth mask of the salient objects is provided for the first frame. The task is thus simplified to only track the objects required. In this paper, the authors look at an unsupervised video object segmentation technique.<br />
<br />
== Background Papers ==<br />
Video object segmentation has been performed using spatio-temporal graphs [[https://pdfs.semanticscholar.org/7221/c3470fa89879aab3ef270570ced15cde28de.pdf 5], [http://ieeexplore.ieee.org/abstract/document/5539893/ 6], [http://openaccess.thecvf.com/content_iccv_2013/papers/Li_Video_Segmentation_by_2013_ICCV_paper.pdf 7], [https://link.springer.com/content/pdf/10.1007/s11263-011-0512-5.pdf 8]] and deep learning. The graph based methods construct 3D spatio-temporal graphs in order to model the inter and the intra-frame relationship of pixels or superpixels in a video. Hence they are computationally slower than deep learning methods and are unable to run at real-time. There are 2 main deep learning techniques for semi-supervised video object segmentation: One Shot Video Object Segmentation (OSVOS) and Learning Video Object Segmentation from Static Images (MaskTrack). Following is a brief description of the new techniques introduced by these papers for semi-supervised video object segmentation task.<br />
<br />
=== OSVOS (One-Shot Video Object Segmentation) ===<br />
<br />
[[File:OSVOS.jpg | 1000px]]<br />
<br />
This paper introduces the technique of using a frame-by-frame object segmentation without any temporal information from the previous frames of the video. The paper uses a VGG-16 network with pre-trained weights from image classification task. This network is then converted into a fully-connected network (FCN) by removing the fully connected dense layers at the end and adding convolution layers to generate a segment mask of the input. This network is then trained on the DAVIS 2016 dataset.<br />
<br />
During testing, the trained VGG-16 FCN is fine-tuned using the first frame of the video using the ground truth. Because this is a semi-supervised case, the segmented mask (ground truth) for the first frame is available. The first frame data is augmented by zooming/rotating/flipping the first frame and the associated segment mask.<br />
<br />
=== MaskTrack (Learning Video Object Segmentation from Static Images) ===<br />
<br />
[[File:MaskTrack.jpg | 500px]]<br />
<br />
MaskTrack takes the output of the previous frame to improve its predictions and to generate the segmentation mask for the next frame. Thus the input to the network is 4 channel wide (3 RGB channels from the frame at time <math>t</math> plus one binary segmentation mask from frame <math>t-1</math>). The output of the network is the binary segmentation mask for frame at time <math>t</math>. Using the binary segmentation mask (referred to as guided object segmentation in the paper), the network is able to use some temporal information from the previous frame to improve its segmentation mask prediction for the next frame.<br />
<br />
The model of the MaskTrack network is similar to a modular VGG-16 and is referred to as MaskTrack ConvNet in the paper. The network is trained offline on saliency segmentation datasets: ECSSD, MSRA 10K, SOD and PASCAL-S. The input mask for the binary segmentation mask channel is generated via non-rigid deformation and affine transformation of the ground truth segmentation mask. Similar data-augmentation techniques are also used during online training. Just like OSVOS, MaskTrack uses the first frame as ground truth (with augmented images) to fine-tune the network to improve prediction score for the particular video sequence.<br />
<br />
A parallel ConvNet network is used to generate a predicted segment mask based on the optical flow magnitude. The optical flow between 2 frames is calculated using the EpicFlow algorithm. The output of the two networks is combined using an averaging operation to generate the final predicted segmented mask.<br />
<br />
Table 1 below gives a summary comparison of the different state of the art algorithms. The noteworthy information included in this table is that the technique presented in this paper is the only one which takes into account long-term temporal information. This is accomplished with a recurrent neural net. Furthermore, the bounding box is also estimated instead of just a segmentation mask. The authors claim that this allows the incorporation of a location prior from the tracked object.<br />
<br />
[[File:Paper19-SegmentationComp.png]]<br />
<br />
== Dataset ==<br />
The three major datasets used in this paper are DAVIS-2016, DAVIS-2017 and Segtrack v2. DAVIS-2016 dataset provides video sequences with only one segment mask for all salient objects. DAVIS-2017 improves the ground truth data by providing segmentation mask for each salient object as a separate color segment mask. Segtrack v2 also provides multiple segmentation mask for all salient objects in the video sequence. These datasets try to recreate real-life scenarios like occlusions, low resolution videos, background clutter, motion blur, fast motion etc.<br />
<br />
== MaskRNN: Introduction ==<br />
Most techniques mentioned above don’t work directly on instance level segmentation of the objects through the video sequence. The above approaches focus on image segmentation on each frame and using additional information (mask propagation and optical flow) from the preceding frame perform predictions for the current frame. To address the instance level segmentation problem, MaskRNN proposes a framework where the salient objects are tracked and segmented by capturing the temporal information in the video sequence using a recurrent neural network.<br />
<br />
== MaskRNN: Overview ==<br />
In a video sequence <math>I = \{I_1, I_2, …, I_T\}</math>, the sequence of <math>T</math> frames are given as input to the network, where the video sequence contains <math>N</math> salient objects. The ground truth for the first frame <math>y_1^*</math> is also provided for <math>N</math> salient objects.<br />
In this paper, the problem is formulated as a time dependency problem and using a recurrent neural network, the prediction of the previous frame influences the prediction of the next frame. The approach also computes the optical flow between frames (optical flow is the apparent motion of objects between two consecutive frames in the form of a 2D vector field representing the displacement in brightness patterns for each pixel, apparent because it depends on the relative motion between the observer and the scene) and uses that as the input to the neural network. The optical flow is also used to align the output of the predicted mask. “The warped prediction, the optical flow itself, and the appearance of the current frame are then used as input for <math>N</math> deep nets, one for each of the <math>N</math> objects.”[1 - MaskRNN] Each deep net is a made of an object localization network and a binary segmentation network. The binary segmentation network is used to generate the segmentation mask for an object. The object localization network is used to alleviate outliers from the predictions. The final prediction of the segmentation mask is generated by merging the predictions of the 2 networks. For <math>N</math> objects, there are N deep nets which predict the mask for each salient object. The predictions are then merged into a single prediction using an <math>\text{argmax}</math> operation at test time.<br />
<br />
== MaskRNN: Multiple Instance Level Segmentation ==<br />
<br />
[[File:2ObjectSeg.jpg | 850px]]<br />
<br />
Image segmentation requires producing a pixel level segmentation mask and this can become a multi-class problem. Instead, using the approach from [2- Mask R-CNN] this approach is converted into a multiple binary segmentation problem. A separate segmentation mask is predicted separately for each salient object and thus we get a binary segmentation problem. The binary segments are combined using an <math>\text{argmax}</math> operation where each pixel is assigned to the object containing the largest predicted probability.<br />
<br />
=== MaskRNN: Binary Segmentation Network ===<br />
<br />
[[File:MaskRNNDeepNet.jpg | 850px]]<br />
<br />
The above picture shows a single deep net employed for predicting the segment mask for one salient object in the video frame. The network consists of 2 networks: binary segmentation network and object localization network. The binary segmentation network is split into two streams: appearance and flow stream. The input of the appearance stream is the RGB frame at time t and the wrapped prediction of the binary segmentation mask from time <math>t-1</math>. The wrapping function uses the optical flow between frame <math>t-1</math> and frame <math>t</math> to generate a new binary segmentation mask for frame <math>t</math>. The input to the flow stream is the concatenation of the optical flow magnitude between frames <math>t-1</math> to <math>t</math> and frames <math>t</math> to <math>t+1</math> and the wrapped prediction of the segmentation mask from frame <math>t-1</math>. The magnitude of the optical flow is replicated into an RBG format before feeding it to the flow stream. The network architecture closely resembles a VGG-16 network without the pooling or fully connected layers at the end. The fully connected layers are replaced with convolutional and bilinear interpolation upsampling layers which are then linearly combined to form a feature representation that is the same size of the input image. This feature representation is then used to generate a binary segment mask. This technique is borrowed from the Fully Convolutional Network mentioned above. The output of the flow stream and the appearance stream is linearly combined and sigmoid function is applied to the result to generate binary mask for ith object. All parts of the network are fully differentiable and thus it can be fully trained in every pass.<br />
<br />
=== MaskRNN: Object Localization Network: ===<br />
Using a similar technique to the Fast-RCNN method of object localization, where the region of interest (RoI) pooling of the features of the region proposals (i.e. the bounding box proposals here) is performed and passed through fully connected layers to perform regression, the Object localization network generates a bounding box of the salient object in the frame. This bounding box is enlarged by a factor of 1.25 and combined with the output of binary segmentation mask. Only the segment mask available in the bounding box is used for prediction and the pixels outside of the bounding box are marked as zero. MaskRNN uses the convolutional feature output of the appearance stream as the input to the RoI-pooling layer to generate the predicted bounding box. A pixel is classified as foreground if it is both predicted to be in the foreground by the binary segmentation net and within the enlarged estimated bounding box from the object localization net.<br />
<br />
=== Training and Finetuning ===<br />
For training the network depicted in Figure 1, backpropagation through time is used in order to preserve the recurrence relationship connecting the frames of the video sequence. Predictive performance is further improved by following the algorithm for semi-supervised setting for video object segmentation with fine-tuning achieved by using the first frame segmentation mask of the ground truth. In this way, the network is further optimized using the ground truth data.<br />
<br />
== MaskRNN: Implementation Details ==<br />
=== Offline Training ===<br />
The deep net is first trained offline on a set of static images. The ground truth is randomly perturbed locally to generate the imperfect mask from frame <math>t-1</math>. Two different networks are trained offline separately for DAVIS-2016 and DAVIS-2017 datasets for a fair evaluation of both datasets. After both the object localization net and binary segmentation networks have trained, the temporal information in the network is used to further improve the segmented prediction results. Because of GPU memory constraints, the RNN is only able to backpropagate the gradients back 7 frames and learn long-term temporal information. <br />
<br />
For optical flow, a pre-trained flowNet2.0 is used to compute the optical flow between frames. (A flowNet (Dosovitskiy 2015) is a deep neural network trained to predict optical flow. The simplest form of flowNet has an architecture consisting of two parts. The first part accepts the two images between which the optical flow is to be computed as input, as applies a sequence of convolution and max-pooling operations, as in a standard convolutional neural network. In the second part, repeated up-convolution operations are applied, increasing the dimensions of the feature-maps. Besides the output of the previous upconvolution, each upconvolution is also fed as input the output of the corresponding down-convolution from the first part of the network. Thus part of the architecture resembles that of a U-net (Ronneberger, 2015). The output of the network is the predicted optical flow. ) <br />
<br />
=== Online Finetuning ===<br />
The deep nets (without the RNN) are then fine-tuned during test time by online training the networks on the ground truth of the first frame and some augmentations of the first frame data. The learning rate is set to <math>10^{-5}</math> for online training for 200 iterations and the learning rate is gradually decayed over time. Data augmentation techniques similar to those in offline training, namely random resizing, rotating, cropping and flipping is applied. Also, it should be noted that the RNN is ''not'' employed during online finetuning since only a single frame of training data is available.<br />
<br />
== MaskRNN: Experimental Results ==<br />
=== Evaluation Metrics ===<br />
There are 3 different techniques for performance analysis for Video Object Segmentation techniques:<br />
<br />
1. Region Similarity (Jaccard Index): Region similarity or Intersection-over-union is used to capture precision of the area covered by the prediction segmentation mask compared to the ground truth segmentation mask.<br />
<br />
\begin{equation}<br />
IoU = \frac{|M \cap G|}{|M| + |G| - |M \cap G|} <br />
\label{equation:Jaccard}<br />
\end{equation}<br />
<br />
2. Contour Accuracy (F-score): This metric measures the accuracy in the boundary of the predicted segment mask and the ground truth segment mask using bipartite matching between the bounding pixels of the masks. <br />
<br />
[[File:Fscore.jpg | 200px]]<br />
<br />
3. Temporal Stability : This estimates the degree of deformation needed to transform the segmentation masks from one frame to the next and is measured by the dissimilarity of the set of points on the contours of the segmentation between two adjacent frames.<br />
<br />
Region similarity measures the true segmented area in the prediction, while Contour Accuracy measures the accuracy of the contours/segmented mask boundary.<br />
<br />
=== Ablation Study ===<br />
<br />
The ablation study summarized how the different components contributed to the algorithm evaluated on DAVIS-2016 and DAVIS-2017 datasets.<br />
<br />
[[File:MaskRNNTable2.jpg | 700px]]<br />
<br />
The above table presents the contribution of each component of the network to the final prediction score. Online fine-tuning improves the performance by a large margin, as the network becomes adjusted to the appearance of the specific object being tracked. Addition of RNN/Localization Net and FStream all seem to positively affect the performance of the deep net.<br />
<br />
=== Quantitative Evaluation ===<br />
<br />
The authors use DAVIS-2016, DAVIS-2017 and Segtrack v2 to compare the performance of the proposed approach to other methods based on foreground-background video object segmentation and multiple instance-level video object segmentation.<br />
<br />
[[File:MaskRNNTable3.jpg | 700px]]<br />
<br />
The above table shows the results for contour accuracy mean and region similarity. The MaskRNN method seems to outperform all previously proposed methods. The performance gain is significant by employing a Recurrent Neural Network for learning recurrence relationship and using a object localization network to improve prediction results.<br />
<br />
The following table shows the improvements in the state of the art achieved by MaskRNN on the DAVIS-2017 and the SegTrack v2 dataset.<br />
<br />
[[File:MaskRNNTable4.jpg | 700px]]<br />
<br />
=== Qualitative Evaluation ===<br />
The authors showed example qualitative results from the DAVIS and Segtrack datasets. <br />
<br />
Below are some success cases of object segmentation under complex motion, cluttered background, and/or multiple object occlusion.<br />
<br />
[[File:maskrnn_example.png | 700px]]<br />
<br />
Below are a few failure cases. The authors explain two reasons for failure: a) when similar objects of interest are contained in the frame (left two images), and b) when there are large variations in scale and viewpoint (right two images).<br />
<br />
[[File:maskrnn_example_fail.png | 700px]]<br />
<br />
== Conclusion ==<br />
In this paper a novel approach to instance level video object segmentation task is presented which performs better than current state of the art. The long-term recurrence relationship is learnt using an RNN. The object localization network is added to improve accuracy of the system. Using online fine-tuning the network is adjusted to predict better for the current video sequence.<br />
<br />
== Critique ==<br />
The paper provides a technique to track multiple objects in a video. The novelty is to add back-propagation through time to improve the tracking accuracy and using a localization network to remove any outliers in the segmented binary mask. However, the network architecture it too large and isn't able to run in real-time. There are N deep-Nets for N objects and each deep-Net contains 2 parallel VGG-16 convolutional networks.<br />
<br />
== Implementation ==<br />
<br />
The implementation of this paper was produced as part of the NIPS Paper Implementation Challenge. This implementation can be found at the following open source project: https://github.com/philferriere/tfvos.<br />
<br />
== References ==<br />
# Dosovitskiy, Alexey, et al. "Flownet: Learning optical flow with convolutional networks." Proceedings of the IEEE International Conference on Computer Vision. 2015.<br />
# Hu, Y., Huang, J., & Schwing, A. "MaskRNN: Instance level video object segmentation". Conference on Neural Information Processing Systems (NIPS). 2017<br />
# Ferriere, P. (n.d.). Semi-Supervised Video Object Segmentation (VOS) with Tensorflow. Retrieved March 20, 2018, from https://github.com/philferriere/tfvos<br />
# Ronneberger, Olaf, Philipp Fischer, and Thomas Brox. "U-net: Convolutional networks for biomedical image segmentation." International Conference on Medical image computing and computer-assisted intervention. Springer, Cham, 2015.<br />
# Lee, Yong Jae, Jaechul Kim, and Kristen Grauman. "Key-segments for video object segmentation." Computer Vision (ICCV), 2011 IEEE International Conference on. IEEE, 2011.<br />
# Grundmann, Matthias, et al. "Efficient hierarchical graph-based video segmentation." Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on. IEEE, 2010.<br />
# Li, Fuxin, et al. "Video segmentation by tracking many figure-ground segments." Computer Vision (ICCV), 2013 IEEE International Conference on. IEEE, 2013.<br />
# Tsai, David, et al. "Motion coherent tracking using multi-label MRF optimization." International journal of computer vision 100.2 (2012): 190-202.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Label-Free_Supervision_of_Neural_Networks_with_Physics_and_Domain_Knowledge&diff=36045Label-Free Supervision of Neural Networks with Physics and Domain Knowledge2018-04-03T03:47:06Z<p>J3ku: Update introduction</p>
<hr />
<div>== Introduction ==<br />
The requirement of large amounts of labeled training data limits the applications of machine learning. Neural networks, in particular, require large amounts of labeled data to work (LeCun, Bengio, and Hinton 2015[1]). Humans are often able to instead learn from high level instructions for how a task should be performed, or what the final result should look like. This work explores whether a similar principle can be applied to teaching machines: can we supervise networks without individual examples by instead describing only the structure of desired outputs?<br />
<br />
[[File:c433li-1.png|300px|center]]<br />
<br />
Unsupervised learning methods such as autoencoders, also aim to uncover hidden structure in the data without having access to any label. Such systems succeed in producing highly compressed, yet informative representations of the inputs (Kingma and Welling 2013; Le 2013). However, these representations differ from ours as they are not explicitly constrained to have a particular meaning or semantics. This paper attempts to explicitly provide the semantics of the hidden variables we hope to discover, but still train without labels by learning from constraints that are known to hold according to prior domain knowledge. By training without direct examples of the values our hidden (output) variables take, several advantages are gained over traditional supervised learning, including:<br />
* a reduction in the amount of work spent labeling, <br />
* an increase in generality, as a single set of constraints can be applied to multiple data sets without relabeling.<br />
<br />
The primary contribution in the paper is to demonstrate how constraint learning can be used to train neural networks, and to explore how to learn useful feature representations from raw data while avoiding trivial, low entropy solutions.<br />
<br />
== Problem Setup ==<br />
In a traditional supervised learning setting, we are given a training set <math>D=\{(x_1, y_1), \cdots, (x_n, y_n)\}</math> of <math>n</math> training examples. Each example is a pair <math>(x_i,y_i)</math> formed by an instance <math>x_i \in X</math> and the corresponding output (label) <math>y_i \in Y</math>. The goal is to learn a function <math>f: X \rightarrow Y</math> mapping inputs to outputs. To quantify performance, a loss function <math>\ell:Y \times Y \rightarrow \mathbb{R}</math> is provided, and a mapping is found via <br />
<br />
<center><math> f^* = \text{argmin}_{f \in \mathcal{F}} \sum_{i=1}^n \ell(f(x_i),y_i) </math></center><br />
<br />
where the optimization is over a pre-defined class of functions <math>\mathcal{F}</math> (hypothesis class). In our case, <math>\mathcal{F}</math> will be (convolutional) neural networks parameterized by their weights. The loss could be for example <math>\ell(f(x_i),y_i) = 1[f(x_i) \neq y_i]</math>. By restricting the space of possible functions specifying the hypothesis class <math>\mathcal{F}</math>, we are leveraging prior knowledge about the specific problem we are trying to solve. Informally, the so-called No Free Lunch Theorems state that every machine learning algorithm must make such assumptions in order to work. Another common way in which a modeler incorporates prior knowledge is by specifying an a-priori preference for certain functions in <math>\mathcal{F}</math>, incorporating a regularization term <math>R:\mathcal{F} \rightarrow \mathbb{R}</math>, and solving for <math> f^* = argmin_{f \in \mathcal{F}} \sum_{i=1}^n \ell(f(x_i),y_i) + R(f)</math>. Typically, the regularization term <math>R:\mathcal{F} \rightarrow \mathbb{R}</math> specifies a preference for "simpler" functions (Occam's razor) to prevent overfitting the model on the training data.<br />
<br />
The focus is on the set of problems/domains where the problem is a complex environment having a complex representation of the output space, for example mapping an input image to the height of an object(since this leads to a complex output space) rather than simple binary classification problem.<br />
<br />
In this paper, prior knowledge on the structure of the outputs is modelled by providing a weighted constraint function <math>g:X \times Y \rightarrow \mathbb{R}</math>, used to penalize “structures” that are not consistent with our prior knowledge. And whether this weak form of supervision is sufficient to learn interesting functions is explored. While one clearly needs labels <math>y</math> to evaluate <math>f^*</math>, labels may not be necessary to discover <math>f^*</math>. If prior knowledge informs us that outputs of <math>f^*</math> have other unique properties among functions in <math>\mathcal{F}</math>, we may use these properties for training rather than direct examples <math>y</math>. <br />
<br />
Specifically, an unsupervised approach where the labels <math>y_i</math> are not provided to us is considered, where a necessary property of the output <math>g</math> is optimized instead.<br />
<center><math>\hat{f}^* = \text{argmin}_{f \in \mathcal{F}} \sum_{i=1}^n g(x_i,f(x_i))+ R(f) </math></center><br />
<br />
If the optimizing the above equation is sufficient to find <math>\hat{f}^*</math>, we can use it in replace of labels. If it's not sufficient, additional regularization terms are added. The idea is illustrated with three examples, as described in the next section.<br />
<br />
== Experiments ==<br />
=== Tracking an object in free fall ===<br />
In the first experiment, they record videos of an object being thrown across the field of view, and aim to learn the object's height in each frame. The goal is to obtain a regression network mapping from <math>{R^{\text{height} \times \text{width} \times 3}} \rightarrow \mathbb{R}</math>, where <math>\text{height}</math> and <math>\text{width}</math> are the number of vertical and horizontal pixels per frame, and each pixel has 3 color channels. This network is trained as a structured prediction problem operating on a sequence of <math>N</math> images to produce a sequence of <math>N</math> heights, <math>\left(R^{\text{height} \times \text{width} \times 3} \right)^N \rightarrow \mathbb{R}^N</math>, and each piece of data <math>x_i</math> will be a vector of images, <math>\mathbf{x}</math>.<br />
Rather than supervising the network with direct labels, <math>\mathbf{y} \in \mathbb{R}^N</math>, the network is instead supervised to find an object obeying the elementary physics of free falling objects. An object acting under gravity will have a fixed acceleration of <math>a = -9.8 m / s^2</math>, and the plot of the object's height over time will form a parabola:<br />
<center><math>\mathbf{y}_i = y_0 + v_0(i\Delta t) + \frac{1}{2} a(i\Delta t)^2</math></center><br />
<br />
The idea is, given any trajectory of <math>N</math> height predictions, <math>f(\mathbf{x})</math>, we fit a parabola with fixed curvature to those predictions, and minimize the resulting residual. Formally, if we specify <math>\mathbf{a} = [\frac{1}{2} a\Delta t^2, \frac{1}{2} a(2 \Delta t)^2, \ldots, \frac{1}{2} a(N \Delta t)^2]</math>, the prediction produced by the fitted parabola is:<br />
<center><math> \text{argmin}_{v_0, y_0}\sum_i(y_i-y_0-v_0(i\Delta_t)-\frac{1}{2}a(i\Delta_t)^2) </math></center><br />
By the solution of ordinary least square estimation: <br />
<center><math> \mathbf{\hat{y}} = \mathbf{a} + \mathbf{A} (\mathbf{A}^T\mathbf{A})^{-1} \mathbf{A}^T (f(\mathbf{x}) - \mathbf{a}) </math></center><br />
<br />
where<br />
<center><br />
<math><br />
\mathbf{A} = <br />
\left[ {\begin{array}{*{20}c}<br />
\Delta t & 1 \\<br />
2\Delta t & 1 \\<br />
3\Delta t & 1 \\<br />
\vdots & \vdots \\<br />
N\Delta t & 1 \\<br />
\end{array} } \right]<br />
</math><br />
</center><br />
<br />
The constraint loss is then defined as<br />
<center><math>g(\mathbf{x},f(\mathbf{x})) = g(f(\mathbf{x})) = \sum_{i=1}^{N} |\mathbf{\hat{y}}_i - f(\mathbf{x})_i|</math></center><br />
<br />
Note that <math>\hat{y}</math> is not the ground truth labels. Because <math>g</math> is differentiable almost everywhere, it can be optimized with SGD. They find that when combined with existing regularization methods for neural networks, this optimization is sufficient to recover <math>f^*</math> up to an additive constant <math>C</math> (specifying what object height corresponds to 0).<br />
<br />
[[File:c433li-2.png|650px|center]]<br />
<br />
The data set is collected on a laptop webcam running at 10 frames per second (<math>\Delta t = 0.1s</math>). The camera position is fixed and 65 diverse trajectories of the object in flight, totalling 602 images are recorded. For each trajectory, the network is trained on randomly selected intervals of <math>N=5</math> contiguous frames. Images are resized to <math>56 \times 56</math> pixels before going into a small, randomly initialized neural network with no pretraining. The network consists of 3 Conv/ReLU/MaxPool blocks followed by 2 Fully Connected/ReLU layers with probability 0.5 dropout and a single regression output.<br />
<br />
Since scaling the <math>y_0</math> and <math>v_0</math> results in the same constraint loss <math>g</math>, the authors evaluate the result by the correlation of predicted heights with ground truth pixel measurements. This method was used since the distance from the object to the camera could not be accurately recorded, and this distance is required to calculate the height in meters. This is not a bullet proof evaluation, and is discussed in further detail in the critique section. The results are compared to a supervised network trained with the labels to directly predict the height of the object in pixels. The supervised learning task is viewed as a substantially easier task. From this knowledge we can see from the table below that, under their evaluation criteria, the result is pretty satisfying.<br />
<br />
==== Evaluation ====<br />
{| class="wikitable"<br />
|-<br />
! scope="col" | Method !! scope="col" | Random Uniform Output !! scope="col" | Supervised with Labels !! scope="col" | Approach in this Paper<br />
|-<br />
! scope="row" | Correlation <br />
| 12.1% || 94.5% || 90.1%<br />
|}<br />
<br />
=== Tracking the position of a walking man ===<br />
In the second experiment, they aim to detect the horizontal position of a person walking across a frame without providing direct labels <math>y \in \mathbb{R}</math> by exploiting the assumption that the person will be walking at a constant velocity over short periods of time. This is formulated as a structured prediction problem <math>f: \left(R^{\text{height} \times \text{width} \times 3} \right)^N \rightarrow \mathbb{R}^N</math>, and each training instances <math>x_i</math> are a vector of images, <math>\mathbf{x}</math>, being mapped to a sequence of predictions, <math>\mathbf{y}</math>. Given the similarities to the first experiment with free falling objects, we might hope to simply remove the gravity term from equation and retrain. However, in this case, that is not possible, as the constraint provides a necessary, but not sufficient, condition for convergence.<br />
<br />
Given any sequence of correct outputs, <math>(\mathbf{y}_1, \ldots, \mathbf{y}_N)</math>, the modified sequence, <math>(\lambda * \mathbf{y}_1 + C, \ldots, \lambda * \mathbf{y}_N + C)</math> (<math>\lambda, C \in \mathbb{R}</math>) will also satisfy the constant velocity constraint. In the worst case, when <math>\lambda = 0</math>, <math>f \equiv C</math>, and the network can satisfy the constraint while having no dependence on the image. The trivial output is avoided by adding two two additional loss terms.<br />
<br />
<center><math>h_1(\mathbf{x}) = -\text{std}(f(\mathbf{x}))</math></center><br />
which seeks to maximize the standard deviation of the output, and<br />
<br />
<center><br />
<math>\begin{split}<br />
h_2(\mathbf{x}) = \hphantom{'} & \text{max}(\text{ReLU}(f(\mathbf{x}) - 10)) \hphantom{\text{ }}+ \\<br />
& \text{max}(\text{ReLU}(0 - f(\mathbf{x})))<br />
\end{split}<br />
</math><br />
</center><br />
which limit the output to a fixed ranged <math>[0, 10]</math>, the final loss is thus:<br />
<br />
<center><br />
<math><br />
\begin{split}<br />
g(\mathbf{x}) = \hphantom{'} & ||(\mathbf{A} (\mathbf{A}^T\mathbf{A})^{-1} \mathbf{A}^T - \mathbf{I}) * f(\mathbf{x})||_1 \hphantom{\text{ }}+ \\<br />
& \gamma_1 * h_1(\mathbf{x}) <br />
\hphantom{\text{ }}+ \\<br />
& \gamma_2 * h_2(\mathbf{x})<br />
% h_2(y) & = \text{max}(\text{ReLU}(y - 10)) + \\<br />
% & \hphantom{=}\hphantom{a} \text{max}(\text{ReLU}(0 - y))<br />
\end{split}<br />
</math><br />
</center><br />
<br />
[[File:c433li-3.png|650px|center]]<br />
<br />
The data set contains 11 trajectories across 6 distinct scenes, totalling 507 images resized to <math>56 \times 56</math>. The network is trained to output linearly consistent positions on 5 strided frames from the first half of each trajectory, and is evaluated on the second half. The boundary violation penalty is set to <math>\gamma_2 = 0.8</math> and the standard deviation bonus is set to <math>\gamma_1 = 0.6</math>.<br />
<br />
As in the previous experiment, the result is evaluated by the correlation with the ground truth. The result is as follows:<br />
==== Evaluation ====<br />
{| class="wikitable"<br />
|-<br />
! scope="col" | Method !! scope="col" | Random Uniform Output !! scope="col" | Supervised with Labels !! scope="col" | Approach in this Paper<br />
|-<br />
! scope="row" | Correlation <br />
| 45.9% || 80.5% || 95.4%<br />
|}<br />
Surprisingly, the approach in this paper beats the same network trained with direct labeled supervision on the test set, which can be attributed to overfitting on the small amount of training data available (as correlation on training data reached 99.8%).<br />
<br />
=== Detecting objects with causal relationships ===<br />
In the previous experiments, the authors explored options for incorporating constraints pertaining to dynamics equations in real-world phenomena, i.e., prior knowledge derived from elementary physics. In this experiment, the authors explore the possibilities of learning from logical constraints imposed on single images. More specifically, they ask whether it is possible to learn from causal phenomena.<br />
<br />
[[File:paper18_Experiment_3.png|400px|center]]<br />
<br />
Here, the authors provide images containing a stochastic collection of up to four characters: Peach, Mario, Yoshi, and Bowser, with each character having small appearance changes across frames due to rotation and reflection. Example images can be seen in Fig. (4). While the existence of objects in each frame is non-deterministic, the generating distribution encodes the underlying phenomenon that Mario will always appear whenever Peach appears. The aim is to create a pair of neural networks <math>f_1, f_2</math> for identifying Peach and Mario, respectively. The networks, <math>f_k : R^{height×width×3} → \{0, 1\}</math>, map the image to the discrete boolean variables, <math>y_1</math> and <math>y_2</math>. Rather than supervising with direct labels, the authors train the networks by constraining their outputs to have the logical relationship <math>y_1 ⇒ y_2</math>. This problem is challenging because the networks must simultaneously learn to recognize the characters and select them according to logical relationships. To avoid the trivial solution <math>y_1 \equiv 1, y_2 \equiv 1</math> on every image, three additional loss terms need to be added:<br />
<br />
<center><math> h_1(\mathbf{x}, k) = \frac{1}{M}\sum_i^M |Pr[f_k(\mathbf{x}) = 1] - Pr[f_k(\rho(\mathbf{x})) = 1]|, </math></center><br />
<br />
which forces rotational independence of the outputs in order to encourage the network to learn the existence, rather than location of objects, <br />
<br />
<center><math> h_2(\mathbf{x}, k) = -\text{std}_{i \in [1 \dots M]}(Pr[f_k(\mathbf{x}_i) = 1]), </math></center><br />
<br />
which seeks high variance outputs, and<br />
<br />
<center><br />
<math> h_3(\mathbf{x}, v) = \frac{1}{M}\sum_i^{M} (Pr[f(\mathbf{x}_i) = v] - \frac{1}{3} + (\frac{1}{3} - \mu_v))^2 \\<br />
\mu_{v} = \frac{1}{M}\sum_i^{M} \mathbb{1}\{v = \text{argmax}_{v' \in \{0, 1\}^2} Pr[f(\mathbf{x}) = v']\}. </math><br />
</center><br />
<br />
which seeks high entropy outputs. The final loss function then becomes: <br />
<br />
<center><br />
<math> \begin{split}<br />
g(\mathbf{x}) & = \mathbb{1}\{f_1(\mathbf{x}) \nRightarrow f_2(\mathbf{x})\} \hphantom{\text{ }} + \\<br />
& \sum_{k \in \{1, 2\}} \gamma_1 h_1(\mathbf{x}, k) + \gamma_2 h_2(\mathbf{x}, k) + <br />
\hspace{-0.7em} \sum_{v \neq \{1,0\}} \hspace{-0.7em} \gamma_3 * h_3(\mathbf{x}, v)<br />
\end{split}<br />
</math><br />
</center><br />
<br />
====Evaluation====<br />
<br />
The input images, shown in Fig. (4), are 56 × 56 pixels. The authors used <math>\gamma_1 = 0.65, \gamma_2 = 0.65, \gamma_3 = 0.95</math>, and trained for 4,000 iterations. This experiment demonstrates that networks can learn from constraints that operate over discrete sets with potentially complex logical rules. Removing constraints will cause learning to fail. Thus, the experiment also shows that sophisticated sufficiency conditions can be key to success when learning from constraints.<br />
<br />
== Conclusion and Critique ==<br />
This paper has introduced a method for using physics and other domain constraints to supervise neural networks. However, the approach described in this paper is not entirely new. Similar ideas are already widely used in Q learning, where the Q value are not available, and the network is supervised by the constraint, as in Deep Q learning (Mnih, Riedmiller et al. 2013[2]).<br />
<center><math>Q(s,a) = R(r,s) + \gamma \sum_{s' ~ P_{sa}}{\text{max}_{a'}Q(s',a')}</math></center><br />
<br />
<br />
Also, the paper has a mistake where they quote the free fall equation as<br />
<center><math>\mathbf{y}_i = y_0 + v_0(i\Delta t) + a(i\Delta t)^2</math></center><br />
which should be<br />
<center><math>\mathbf{y}_i = y_0 + v_0(i\Delta t) + \frac{1}{2} a(i\Delta t)^2</math></center><br />
Although in this case it doesn't affect the result.<br />
<br />
<br />
For the evaluation of the experiments, they used correlation with ground truth as the metric to avoid the fact that the output can be scaled without affecting the constraint loss. This is fine if the network gives output of the same scale. However, there's no such guarantee, and the network may give output of varying scale for different inputs, in which case, we can't say that the network has learnt the correct thing, although it may have a high correlation with ground truth. In fact, to solve the scaling issue, an obvious way is to combine the constraints introduced in this paper with some labeled training data. It's not clear why the author didn't experiment with a combination of these two losses.<br />
<br />
In regards to the free fall experiment in particular, the authors apply a fixed acceleration model to create the constraint loss, with the goal of having the network predict height. However, since they did not measure the true height of the object to create test labels, they evaluate using height in pixel space. They do not mention the accuracy of their camera calibration, nor what camera model was used to remove lens distortion. Since lens distortion tends to be worse at the extreme edges of the image, and that they tossed the pillow throughout the entire frame, it is likely that the ground truth labels were corrupted by distortion. If that is the case, it is possible the supervised network is actually performing worse, because it learning how to predict distorted (beyond a constant scaling factor) heights instead of the true height.<br />
<br />
These methods essentially boil down to generating approximate labels for training data using some knowledge of the dynamic that the labels should follow.<br />
<br />
Finally, this paper only picks examples where the constraints are easy to design, while in some more common tasks such as image classification, what kind of constraints are needed is not straightforward at all.<br />
<br />
== References ==<br />
[1] LeCun, Y.; Bengio, Y.; and Hinton, G. 2015. Deep learning. Nature 521(7553):436–444.<br />
<br />
[2] Mnih, V.; Kavukcuoglu, K.; Silver, D.; Graves, A.; Antonoglou, I.; Wierstra, D.; and Riedmiller, M. 2013. Playing Atari with Deep Reinforcement Learning. arxiv 1312.5602.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=A_Neural_Representation_of_Sketch_Drawings&diff=36044A Neural Representation of Sketch Drawings2018-04-03T03:34:54Z<p>J3ku: Add listed contribution, add weighted loss experiment info</p>
<hr />
<div>= Introduction =<br />
<br />
There have been many recent advances in neural generative models for low resolution pixel-based images. Humans, however, do not see the world in a grid of pixels and more typically communicate drawings of the things we see using a series of pen strokes that represent components of objects. These pen strokes are similar to the way vector-based images store data. This paper proposes a new method for creating conditional and unconditional generative models for creating these kinds of vector sketch drawings based on recurrent neural networks (RNNs). For the conditional generation mode, the authors explore the model's latent space that it uses to express the vector image. The paper also explores many applications of these kinds of models, especially creative applications and makes available their unique dataset of vector images.<br />
<br />
= Related Work =<br />
<br />
Previous work related to sketch drawing generation includes methods that focused primarily on converting input photographs into equivalent vector line drawings. Image generating models using neural networks also exist but focused more on generation of pixel-based imagery. For example, Gatys et al.'s (2015) work focuses on separating style and content from pixel-based artwork and imagery. Some recent work has focused on handwritten character generation using RNNs and Mixture Density Networks to generate continuous data points. This work has been extended somewhat recently to conditionally and unconditionally generate handwritten vectorized Chinese Kanji characters by modeling them as a series of pen strokes. Furthermore, this paper builds on work that employed Sequence-to-Sequence models with Variational Auto-encoders to model English sentences in latent vector space.<br />
<br />
One of the limiting factors for creating models that operate on vector datasets has been the dearth of publicly available data. Previously available datasets include Sketch, a set of 20K vector drawings; Sketchy, a set of 70K vector drawings; and ShadowDraw, a set of 30K raster images with extracted vector drawings.<br />
<br />
= Methodology =<br />
<br />
=== Dataset ===<br />
<br />
The “QuickDraw” dataset used in this research was assembled from 75K user drawings extracted from the game “Quick, Draw!” where users drew objects from one of hundreds of classes in 20 seconds or less. The dataset is split into 70K training samples and 2.5K validation and test samples each and represents each sketch a set of “pen stroke actions”. Each action is provided as a vector in the form <math>(\Delta x, \Delta y, p_{1}, p_{2}, p_{3})</math>. For each vector, <math>\Delta x</math> and <math>\Delta y</math> give the movement of the pen from the previous point, with the initial location being the origin. The last three vector elements are a one-hot representation of pen states; <math>p_{1}</math> indicates that the pen is down and a line should be drawn between the current point and the next point, <math>p_{2}</math> indicates that the pen is up and no line should be drawn between the current point and the next point, and <math>p_{3}</math> indicates that the drawing is finished and subsequent points and the current point should not be drawn.<br />
<br />
=== Sketch-RNN ===<br />
[[File:sketchrnn.PNG]]<br />
<br />
The model is a Sequence-to-Sequence Variational Autoencoder (VAE). The encoder model is a symmetric and parallel set of two RNNs that individually process the sketch drawings (sequence <math>S</math>) in forward and reverse order, respectively. The hidden state produced by each encoder model is then concatenated into a single hidden state <math>h</math>. <br />
<br />
\begin{align}<br />
h_\rightarrow = \text{encode}_\rightarrow(S), h_\leftarrow = \text{encode}_\leftarrow(S_{\text{reverse}}), h=[h_\rightarrow; h_\leftarrow]<br />
\end{align}<br />
<br />
The concatenated hidden state <math>h</math> is then projected into two vectors <math>\mu</math> and <math>\hat{\sigma}</math> each of size <math>N_{z}</math> using a fully connected layer. <math>\hat{\sigma}</math> is then converted into a non-negative standard deviation parameter <math>\sigma</math> using an exponential operator. These two parameters <math>\mu</math> and <math>\sigma</math> are then used along with an IID Gaussian vector distributed as <math>\mathcal{N}(0, I)</math> of size <math>N_{z}</math> to construct a random vector <math>z \in ℝ^{N_{z}}</math>, similar to the method used for VAE:<br />
\begin{align}<br />
\mu = W_{\mu}h + b_{mu}\textrm{, }\hat{\sigma} = W_{\sigma}h + b_{\sigma}\textrm{, }\sigma = exp\bigg{(}\frac{\hat{\sigma}}{2}\bigg{)}\textrm{, }z = \mu + \sigma \odot \mathcal{N}(0,I)<br />
\end{align}<br />
<br />
The decoder model is an autoregressive RNN that samples output sketches from the latent vector <math>z</math>. The initial hidden states of each recurrent neuron are determined using <math>[h_{0}, c_{0}] = tanh(W_{z}z + b_{z})</math>. Each step of the decoder RNN accepts the previous point <math>S_{i-1}</math> and the latent vector <math>z</math> as concatenated input. The initial point given is the origin point with pen state down. The output at each step are the parameters for a probability distribution of the next point <math>S_{i}</math>. Outputs <math>\Delta x</math> and <math>\Delta y</math> are modeled using a Gaussian Mixture Model (GMM) with M normal distributions and output pen states <math>(q_{1}, q_{2}, q_{3})</math> modelled as a categorical distribution with one-hot encoding.<br />
\begin{align}<br />
P(\Delta x, \Delta y) = \sum_{j=1}^{M}\Pi_{j}\mathcal{N}(\Delta x, \Delta y | \mu_{x, j}, \mu_{y, j}, \sigma_{x, j}, \sigma_{y, j}, \rho_{xy, j})\textrm{, where }\sum_{j=1}^{M}\Pi_{j} = 1<br />
\end{align}<br />
<br />
For each of the M distributions in the GMM, parameters <math>\mu</math> and <math>\sigma</math> are output for both the x and y locations signifying the mean location of the next point and the standard deviation, respectively. Also output from each model is parameter <math>\rho_{xy}</math> signifying correlation of each bivariate normal distribution. An additional vector <math>\Pi</math> is an output giving the mixture weights for the GMM. The output <math>S_{i}</math> is determined from each of the mixture models using softmax sampling from these distributions.<br />
<br />
One of the key difficulties in training this model is the highly imbalanced class distribution of pen states. In particular, the state that signifies a drawing is complete will only appear one time per each sketch and is difficult to incorporate into the model. In order to have the model stop drawing, the authors introduce a hyperparameter <math>N_{max}</math> which basically is the length of the longest sketch in the dataset and limits the number of points per drawing to being no more than <math>N_{max}</math>, after which all output states form the model are set to (0, 0, 0, 0, 1) to force the drawing to stop.<br />
<br />
To sample from the model, the parameters required by the GMM and categorical distributions are generated at each time step and the model is sampled until a “stop drawing” state appears or the time state reaches time <math>N_{max}</math>. The authors also introduce a “temperature” parameter <math>\tau</math> that controls the randomness of the drawings by modifying the pen states, model standard deviations, and mixture weights as follows:<br />
<br />
\begin{align}<br />
\hat{q}_{k} \rightarrow \frac{\hat{q}_{k}}{\tau}\textrm{, }\hat{\Pi}_{k} \rightarrow \frac{\hat{\Pi}_{k}}{\tau}\textrm{, }\sigma^{2}_{x} \rightarrow \sigma^{2}_{x}\tau\textrm{, }\sigma^{2}_{y} \rightarrow \sigma^{2}_{y}\tau<br />
\end{align}<br />
<br />
This parameter <math>\tau</math> lies in the range (0, 1]. As the parameter approaches 0, the model becomes more deterministic and always produces the point locations with the maximum likelihood for a given timestep.<br />
<br />
=== Unconditional Generation ===<br />
<br />
[[File:paper15_Unconditional_Generation.png|800px|]]<br />
<br />
The authors also explored unconditional generation of sketch drawings by only training the decoder RNN module. To do this, the initial hidden states of the RNN were set to 0, and only vectors from the drawing input are used as input without any conditional latent variable <math>z</math>. Figure 3 above shows different sketches that are sampled from the network by only varying the temperature parameter <math>\tau</math> between 0.2 and 0.9.<br />
<br />
=== Training ===<br />
The training procedure follows the same approach as training for VAE and uses a loss function that consists of the sum of Reconstruction Loss <math>L_{R}</math> and KL Divergence Loss <math>L_{KL}</math>. The reconstruction loss term is composed of two terms; <math>L_{s}</math>, which tries to maximize the log-likelihood of the generated probability distribution explaining the training data <math>S</math> and <math>L_{p}</math> which is the log loss of the pen state terms.<br />
\begin{align}<br />
L_{s} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{S}}log\bigg{(}\sum_{j=1}^{M}\Pi_{j,i}\mathcal{N}(\Delta x_{i},\Delta y_{i} | \mu_{x,j,i},\mu_{y,j,i},\sigma_{x,j,i},\sigma_{y,j,i},\rho_{xy,j,i})\bigg{)}<br />
\end{align}<br />
\begin{align}<br />
L_{p} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{max}} \sum_{k=1}^{3}p_{k,i}log(q_{k,i})<br />
\end{align}<br />
\begin{align}<br />
L_{R} = L_{s} + L{p}<br />
\end{align}<br />
<br />
The KL divergence loss <math>L_{KL}</math> measures the difference between the latent vector <math>z</math> and an IID Gaussian distribution with 0 mean and unit variance. This term, normalized by the number of dimensions <math>N_{z}</math> is calculated as:<br />
\begin{align}<br />
L_{KL} = -\frac{1}{2N_{z}}\big{(}1 + \hat{\sigma} - \mu^{2} – exp(\hat{\sigma})\big{)}<br />
\end{align}<br />
<br />
The loss for the entire model is thus the weighted sum:<br />
\begin{align}<br />
Loss = L_{R} + w_{KL}L_{KL}<br />
\end{align}<br />
<br />
The value of the weight parameter <math>w_{KL}</math> has the effect that as <math>w_{KL} \rightarrow 0</math>, there is a loss in ability to enforce a prior over the latent space and the model assumes the form of a pure autoencoder. As with VAEs, there is a trade-off between optimizing for the two loss terms (i.e. between how precisely the model can regenerate training data <math>S</math> and how closely the latent vector <math>z</math> follows a standard normal distribution) - smaller values of <math>w_{KL}</math> lead to better <math>L_R</math> and worse <math>L_{KL}</math> compared to bigger values of <math>w_{KL}</math>. Also for unconditional generation, the model is a standalone decoder, so there will be no <math>L_{KL}</math> term as only <math>L_{R}</math> is optimized for. This trade-off is illustrated in Figure 4 showing different settings of <math>w_{KL}</math> and the resulting <math>L_{KL}</math> and <math>L_{R}</math>, as well as just <math>L_{R}</math> in the case of unconditional generation with only a standalone decoder.<br />
<br />
[[File:paper15_fig4.png|600px]]<br />
<br />
=== Model Configuration ===<br />
In the given model, the encoder and decoder RNNs consist of 512 and 2048 nodes respectively. Also, M = 20 mixture components are used for the decoder RNN. Layer Normalization is applied to the model, and during training recurrent dropout is applied with a keep probability of 90%. The model is trained with batch sizes of 100 samples, using Adam with a learning rate of 0.0001 and gradient clipping of 1.0. During training, simple data augmentation is performed by multiplying the offset columns by two IID random factors. <br />
<br />
= Experiments =<br />
The authors trained multiple conditional and unconditional models using varying values of <math>w_{KL}</math> and recorded the different <math>L_{R}</math> and <math>L_{KL}</math> values at convergence. The network used LSTM as it’s encoder RNN and HyperLSTM as the decoder network. The HyperLSTM model was used for decoding because it has a history of being useful in sequence generation tasks. (A HyperLSTM consists of two coupled LSTMS: an auxiliary LSTM and a main LSTM. At every time step, the auxiliary LSTM reads the previous hidden state and the current input vector, and computes an intermediate vector <math display="inline"> z </math>. The weights of the main LSTM used in the current time step are then a learned function of this intermediate vector <math display="inline"> z </math>. That is, the weights of the main LSTM are allowed to vary between time steps as a function of the output of the auxiliary LSTM. See Ha et al. (2016) for details)<br />
<br />
=== Conditional Reconstruction ===<br />
[[File:conditional_generation.PNG]]<br />
<br />
The authors qualitatively assessed the reconstructed images <math>S’</math> given input sketch <math>S</math> using different values for the temperature hyperparameter <math>\tau</math>. The figure above shows the results for different values of <math>\tau</math> starting with 0.01 at the far left and increasing to 1.0 on the far right. Interestingly, sketches with extra features like a cat with 3 eyes are reproduced as a sketch of a cat with two eyes and sketches of object of a different class such as a toothbrush are reproduced as a sketch of a cat that maintains several of the input toothbrush sketches features.<br />
<br />
=== Latent Space Interpolation ===<br />
[[File:latent_space_interp.PNG]]<br />
<br />
The latent space vectors <math>z</math> have few “gaps” between encoded latent space vectors due to the enforcement of a Gaussian prior. This allowed the authors to do simple arithmetic on the latent vectors from different sketches and produce logical resulting images in the same style as latent space arithmetic on Word2Vec vectors. A model trained with higher w_{KL} is expected to produce images closer to the data manifold, and the figure above shows reconstructed images from latent vector interpolation between the original images. Results from the model trained with higher w_{KL} seem to produce more coherent images.<br />
<br />
=== Sketch Drawing Analogies ===<br />
Given the latent space arithmetic possible, it was found that features of a sketch could be added after some sketch input was encoded. For example, a drawing of a cat with a body could be produced by providing the network with a drawing of a cat’s head, and then adding a latent vector to the embedding layer that represents “body”. As an example, this “body” vector might be produced by taking a drawing of a pig with a body and subtracting a vector representing the pigs head.<br />
<br />
=== Predicting Different Endings of Incomplete Sketches ===<br />
[[File:predicting_endings.PNG]]<br />
<br />
Using the decoder RNN only, it is possible to finish sketches by conditioning future vector line predictions on the previous points. To do this, the decoder RNN is first used to encode some existing points into the hidden state of the decoder network and then generating the remaining points of the sketch with <math>\tau</math> set to 0.8.<br />
<br />
= Applications and Future Work =<br />
Sketch-RNN may enable the production of several creative applications. These might include suggesting ways an artist could finish a sketch, enabling artists to explore latent space arithmetic to find interesting outputs given different sketch inputs, or allowing the production of multiple different sketches of some object as a purely generative application. The authors suggest that providing some conditional sketch of an object to a model designed to produce output from a different class might be useful for producing sketches that morph the two different object classes into one sketch. For example, the image below was trained on drawing cats, but a chair was used as the input. This results in a chair looking cat.<br />
<br />
[[File:cat-chair.png]]<br />
<br />
Sketch-RNN may also be useful as a teaching tool to help people learn how to draw, especially if it were to be trained on higher quality images. Teaching tools might suggest to students how to proceed to finish a sketch or intake low fidelity sketches to produce a higher quality and “more coherent” output sketch.<br />
<br />
The authors noted that Sketch-RNN is not as effective at generating coherent sketches when trained on a large number of classes simultaneously (experiments mostly used datasets consisting of one or two object classes), and plan to use class information outside the latent space to try to model a greater number of classes.<br />
<br />
Finally, the authors suggest that combining this model with another that produces photorealistic pixel-based images using sketch input, such as Pix2Pix may be an interesting direction for future research. In this case, the output from the Sketch-RNN model would be used as input for Pix2Pix and could produce photorealistic images given some crude sketch from a user.<br />
<br />
= Limitations =<br />
The authors note a major limitation to the model is the training time relative to the number of data points. When sketches surpass 300 data points the model is difficult to train. To counteract this effect the Ramer-Douglas-Peucker algorithm was used to reduce the number of data points per sketch. This algorithm attempts to significantly reduce the number of data points while keeping the sketch as close to the original as possible.<br />
<br />
Another limitation is the effectiveness of generating sketches as the complexity of the class increases. Below are sketches of a few classes which show how the less complex classes such as cats and crabs are more accurately generated. Frogs (more complex) tend to have overly smooth lines drawn which do not seem to be part of realistic frog samples.<br />
<br />
[[File:paper15_classcomplexity.png]]<br />
<br />
= Conclusion =<br />
The authors presented Sketch-RNN, a RNN model for modelling and generating vector-based sketch drawings. The VAE inspired architecture allows sampling the latent space to generate new drawings and also allows for applications that use latent space arithmetic in the style of Word2Vec to produce new drawings given operations on embedded sketch vectors. The authors also made available a large dataset of sketch drawings in the hope of encouraging more research in the area of vector-based image modelling.<br />
<br />
= Criticisms =<br />
The paper produces an interesting model that can effectively model vector-based images instead of traditional pixel-based images. This is an interesting problem because vector based images require producing a new way to encode the data. While the results from this paper are interesting, most of the techniques used are borrowed ideas from Variational Autoencoders and the main architecture is not terribly groundbreaking. <br />
<br />
One novel part about the architecture presented was the way the authors used GMMs in the decoder network. While this was interesting and seemed to allow the authors to produce different outputs given the same latent vector input <math>z</math> by manipulating the <math>\tau</math> hyperparameter, it was not that clear in the article why GMMs were used instead of a more simple architecture. Much time was spent explaining basics about GMM parameters like <math>\mu</math> and <math>\sigma</math>, but there was comparatively little explanation about how points were actually sampled from these mixture models.<br />
<br />
Finally, the authors gloss somewhat over how they were able to encode previous sketch points using only the decoder network into the hidden state of the decoder RNN to finish partially finished sketches. I can only assume that some kind of back-propagation was used to encode the expected sketch points into the hidden parameters of the decoder, but no explanation was given in the paper.<br />
<br />
== Major Contributions ==<br />
<br />
* For images composed by sequence of lines, such as hand drawing, this paper proposed a framework to generate such image in vector format, conditionally and unconditionally. <br />
* Provided a unique training procedure that targets vector images, which makes training procedures more robust.<br />
* Composed large dataset of hand drawn vector images which benefits future development.<br />
* Discussed several potential applications of this methodology, such as drawing assist for artists and educational tool for students. <br />
<br />
= Implementation =<br />
Google has released all code related to this paper at the following open source repository: https://github.com/tensorflow/magenta/tree/master/magenta/models/sketch_rnn<br />
<br />
= Source =<br />
<br />
# Ha, D., & Eck, D. A neural representation of sketch drawings. In Proc. International Conference on Learning Representations (2018).<br />
# Tensorflow/magenta. (n.d.). Retrieved March 25, 2018, from https://github.com/tensorflow/magenta/tree/master/magenta/models/sketch_rnn</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Robust_Imitation_of_Diverse_Behaviors&diff=36043Robust Imitation of Diverse Behaviors2018-04-03T03:22:39Z<p>J3ku: Add to pros and cons of VAE and GAIL</p>
<hr />
<div>=Introduction=<br />
One of the longest standing challenges in AI is building versatile embodied agents, both in the form of real robots and animated avatars, capable of a wide and diverse set of behaviors. State-of-the-art robots cannot compete with the effortless variety and adaptive flexibility of motor behaviors produced by toddlers. Towards addressing this challenge, the authors combine several deep generative approaches to imitation learning in a way that accentuates their individual strengths and addresses their limitations. The end product is a robust neural network policy that can imitate a large and diverse set of behaviors using few training demonstrations.<br />
<br />
=Motivation=<br />
Deep generative models have recently shown great promise in imitation learning. The authors primarily talk about two approaches viz. supervised approaches that condition on demonstrations and Generative Adversarial Imitation Learning (GAIL). The authors also talk about the strengths and limitations of the two approaches and try to combine the two approaches in order to get the best of both worlds.<br />
<br />
* Supervised approaches that condition on demonstrations using a variational autoencoder (VAE):<br />
** They require large training datasets in order to work for non-trivial tasks<br />
** Experiments show that the VAE learns a structured semantic embedding space, allowing for smooth policy interpolation<br />
** They tend to be brittle and fail when the agent diverges too much from the demonstration trajectories (As proof of this brittleness, the authors cite Ross et al. (2010), who provide a theorem showing that the cost incurred by this kind of model when it deviates from a demonstration trajectory with a small probability can be amplified in a manner quadratic in the number of time steps.)<br />
* Generative Adversarial Imitation Learning (GAIL)<br />
** Allows learning more robust policies with fewer demonstrations<br />
** Adversarial training leads to mode-collapse (the tendency of adversarial generative models to cover only a subset of modes of a probability distribution, resulting in a failure to produce adequately diverse samples)<br />
** More difficult and slow to train as they do not immediately provide a latent representation of the data<br />
<br />
Thus, the former approach can model diverse behaviors without dropping modes but does not learn robust policies, while the latter approach gives robust policies but insufficiently diverse behaviors. Thus, the authors combine the favorable aspects of these two approaches. The base of their model is a new type of variational autoencoder on demonstration trajectories that learns semantic policy embeddings. Leveraging these policy representations, they develop a new version of GAIL that <br />
# is much more robust than the purely-supervised controller, especially with few demonstrations, and <br />
# avoids mode collapse, capturing many diverse behaviors when GAIL on its own does not.<br />
<br />
=Model=<br />
The authors first introduce a variational autoencoder (VAE) for supervised imitation, consisting of a bi-directional LSTM encoder mapping demonstration sequences to embedding vectors, and two decoders. The first decoder is a multi-layer perceptron (MLP) policy mapping a trajectory embedding and the current state to a continuous action vector. The second is a dynamics model mapping the embedding and previous state to the present state while modeling correlations among states with a WaveNet.<br />
<br />
[[File: Model_Architecture.png|700px|center|]]<br />
<br />
==Behavioral cloning with VAE suited for control==<br />
<br />
In this section, the authors follow a similar approach to Duan et al. (2017), but opt for stochastic VAEs as having a distribution <math display="inline">q_\phi(z|x_{1:T})</math> to better regularize the latent space. In their VAE, an encoder stochastically maps a demonstration sequence to an embedding vector <math display="inline">z</math>. Given <math display="inline">z</math>, they decode both the state and action trajectories as shown in the figure above. To train the model, the following loss is minimized:<br />
<br />
\begin{align}<br />
L\left( \alpha, w, \phi; \tau_i \right) = - \pmb{\mathbb{E}}_{q_{\phi}(z|x_{1:T_i}^i)} \left[ \sum_{t=1}^{T_i} log \pi_\alpha \left( a_t^i|x_t^i, z \right) + log p_w \left( x_{t+1}^i|x_t^i, z\right) \right] +D_{KL}\left( q_\phi(z|x_{1:T_i}^i)||p(z) \right)<br />
\end{align}<br />
<br />
Where <math> \alpha </math> parameterizes the action decoder, <math> w </math> parameterizes the state decoder, <math> \phi </math> parameterizes the state encoder, and <math> T_i \in \tau_i </math> is the set of demonstration trajectories.<br />
<br />
The encoder <math display="inline">q</math> uses a bi-directional LSTM. To produce the final embedding, it calculates the average of all the outputs of the second layer of this LSTM before applying a final linear transformation to<br />
generate the mean and standard deviation of a Gaussian. Then, one sample from this Gaussian is taken as the demonstration encoding.<br />
<br />
The action decoder is an MLP that maps the concatenation of the state and the embedding of the parameters of a Gaussian policy. The state decoder is similar to a conditional WaveNet model. In particular, it conditions on the embedding <math display="inline">z</math> and previous state <math display="inline">x_{t-1}</math> to generate the vector <math display="inline">x_t</math> autoregressively. That is, the autoregression is over the components of the vector <math display="inline">x_t</math>. Finally, instead of a Softmax, the model uses a mixture of Gaussians as the output of the WaveNet.<br />
<br />
==Diverse generative adversarial imitiation learning==<br />
To enable GAIL to produce diverse solutions, the authors condition the discriminator on the embeddings generated by the VAE encoder and integrate out the GAIL objective with respect to the variational posterior <math display="inline">q_\phi(z|x_{1:T})</math>. Specifically, the authors train the discriminator by optimizing the following objective:<br />
<br />
\begin{align}<br />
{max}_{\psi} \pmb{\mathbb{E}}_{\tau_i \sim \pi_E} \left( \pmb{\mathbb{E}}_{q(z|x_{1:T_i}^i)} \left[\frac{1}{T_i} \sum_{t=1}^{T_i} logD_{\psi} \left( x_t^i, a_t^i | z \right) + \pmb{\mathbb{E}}_{\pi_\theta} \left[ log(1 - D_\psi(x, a | z)) \right] \right] \right)<br />
\end{align}<br />
<br />
There is related work which uses a conditional GAIL objective to learn controls for multiple behaviors from state trajectories, but the discriminator conditions on an annotated class label, as in conditional GANs.<br />
<br />
The authors condition on unlabeled trajectories, which have been passed through a powerful encoder, and hence this approach is capable of one-shot imitation learning. Moreover, the VAE encoder enables to obtain a continuous latent embedding space where interpolation is possible.<br />
<br />
Since the discriminator is conditional, the reward function is also conditional and clipped so that it is upper-bounded. Conditioning on <math>z</math> allows for the generation of an infinite number of reward functions, each tailored to imitate a different trajectory. Due to the diversity of the reward functions, the policy gradients will not collapse into one particular mode through mode skewing.<br />
<br />
To better motivate the objective, the authors propose on temporarily leaving the context of imitation learning and considering an alternative objective for training GANs<br />
<br />
\begin{align}<br />
{min}_{G}{max}_{D} V (G, D) = \int_{y} p(y) \int_{z} q(z|y) \left[ log D(y | z) + \int_{\hat{y}} G(\hat{y} | z) log (1 - D(\hat{y} | z)) d\hat{y} \right] dy dz<br />
\end{align}<br />
<br />
This function is a simplification of the previous objective function. Furthermore, it satisfies the following property.<br />
<br />
===Lemma 1===<br />
Assuming that <math display="inline">q</math> computes the true posterior distribution that is <math display="inline">q(z|y) = \frac{p(y|z)p(z)}{p(y)}</math> then<br />
<br />
\begin{align}<br />
V (G, D) = \int_{z} p(z) \left[ \int_{y} p(y|z) log D(y|z) dy + \int_{\hat{y}} G(\hat{y} | z) log (1 - D(\hat{y} | z)) d\hat{y} \right] dz<br />
\end{align}<br />
<br />
If an optimal discriminator is further assumed, the cost optimized by the generator then becomes<br />
<br />
\begin{align}<br />
C(G) = 2 \int_ p p(z) JSD[p(\cdot|z) || G(\cdot|z)] dz - log4<br />
\end{align}<br />
<br />
where <math display="inline">JSD</math> stands for the Jensen-Shannon divergence. In the context of the WaveNet described in the earlier section, <math>p(x)</math> is the distribution of a mixture of Gaussians, and <math>p(z)</math> is the distribution over the mixture components, so the conditional distribution over the latent <math>z</math>, <math>p(x | z)</math> is uni-modal, and optimizing the divergence will not lead to mode collapse.<br />
<br />
==Policy Optimization Strategy: TRPO==<br />
<br />
[[file:robust_behaviour_alg.png | 800px]]<br />
<br />
In Algorithm 1, it states that TRPO is used for policy parameter updates. TRPO is short for Trust Region Policy Optimization, which an iterative procedure for policy optimization, developed by John Schulman, Sergey Levine, Philip Moritz, Micheal Jordan and Pieter Abbeel. This optimization methods achieves monotonic improve in fields related to robotic motions, such as walking and swimming. For more details on TRPO, please refer to the [https://arxiv.org/pdf/1502.05477.pdf original paper].<br />
<br />
=Experiments=<br />
<br />
The primary focus of the paper's experimental evaluation is to demonstrate that the architecture allows learning of robust controllers capable of producing the full spectrum of demonstration behaviors for a diverse range of challenging control problems. The authors consider three bodies: a 9 DoF robotic arm, a 9 DoF planar walker, and a 62 DoF complex humanoid (56-actuated joint angles, and a freely translating and rotating 3d root joint). While for the reaching task BC is sufficient to obtain a working controller, for the other two problems the full learning procedure is critical.<br />
<br />
The authors analyze the resulting embedding spaces and demonstrate that they exhibit a rich and sensible structure that can be exploited for control. Finally, the authors show that the encoder can be used to capture the gist of novel demonstration trajectories which can then be reproduced by the controller.<br />
<br />
==Robotic arm reaching==<br />
In this experiment, the authors demonstrate the effectiveness of their VAE architecture and investigate the nature of the learned embedding space on a reaching task with a simulated Jaco arm.<br />
<br />
To obtain demonstrations, the authors trained 60 independent policies to reach to random target locations in the workspace starting from the same initial configuration. 30 trajectories from each of<br />
the first 50 policies were generated. These served as training data for the VAE model (1500 training trajectories in total). The remaining 10 policies were used to generate test data.<br />
<br />
Here are the trajectories produced by the VAE model.<br />
<br />
[[File: Robotic_arm_reaching_VAE.png|300px|center|]]<br />
<br />
The reaching task is relatively simple, so with this amount of data the VAE policy is fairly robust. After training, the VAE encodes and reproduces the demonstrations as shown in the figure below.<br />
<br />
[[File: Robotic_arm_reaching.png|650px|center|]]<br />
<br />
==2D Walker==<br />
<br />
As a more challenging test compared to the reaching task, the authors consider bipedal locomotion. Here, the authors train 60 neural network policies for a 2d walker to serve as demonstrations. These policies are each trained to move at different speeds both forward and backward depending on a label provided as additional input to the policy. Target speeds for training were chosen from a set of four different speeds (m/s): -1, 0, 1, 3.<br />
<br />
[[File: 2D_Walker.png|650px|center|]]<br />
<br />
The authors trained their model with 20 episodes per policy (1200 demonstration trajectories in total, each with a length of 400 steps or 10s of simulated time). In this experiment a full approach is required: training the VAE with BC alone can imitate some of the trajectories, but it performs poorly in general, presumably because the relatively small training set does not cover the space of trajectories sufficiently densely. On this generated dataset, they also train policies with GAIL using the same architecture and hyper-parameters. Due to the lack of conditioning, GAIL does not reproduce coherently trajectories. Instead, it simply meshes different behaviors together. In addition, the policies trained with GAIL also, exhibit dramatically less diversity; see [https://www.youtube.com/watch?v=kIguLQ4OwuM video].<br />
<br />
[[File: 2D_Walker_Optimized.gif|frame|center|In the left panel, the planar walker demonstrates a particular walking style. In the right panel, the model's agent imitates this walking style using a single policy network.]]<br />
<br />
==Complex humanoid==<br />
For this experiment, the authors consider a humanoid body of high dimensionality that poses a hard control problem. They generate training trajectories with the existing controllers, which can produce instances of one of six different movement styles. Examples of such trajectories are shown in Fig. 5.<br />
<br />
[[File: Complex_humanoid.png|650px|center|]]<br />
<br />
The training set consists of 250 random trajectories from 6 different neural network controllers that were trained to match 6 different movement styles from the CMU motion capture database. Each trajectory is 334 steps or 10s long. The authors use a second set of 5 controllers from which they generate trajectories for evaluation (3 of these policies were trained on the same movement styles as the policies used for generating training data).<br />
<br />
Surprisingly, despite the complexity of the body, supervised learning is quite effective at producing sensible controllers: The VAE policy is reasonably good at imitating the demonstration trajectories, although it lacks the robustness to be practically useful. Adversarial training dramatically improves the stability of the controller. The authors analyze the improvement quantitatively by computing the percentage of the humanoid falling down before the end of an episode while imitating either training or test policies. The results are summarized in Figure 5 right. The figure further shows sequences of frames of representative demonstration and associated imitation trajectories. Videos of demonstration and imitation behaviors can be found in the [https://www.youtube.com/watch?v=NaohsyUxpxw video].<br />
<br />
[[File: Complex_humanoid_optimized.gif|frame|center|In the left and middle panels we show two demonstrated behaviors. In the right panel, the model's agent produces an unseen transition between those behaviors.]]<br />
<br />
Also, for practical purposes, it is desirable to allow the controller to transition from one behavior to another. The authors test this possibility in an experiment similar to the one for the Jaco arm: They determine the<br />
embedding vectors of pairs of demonstration trajectories, start the trajectory by conditioning on the first embedding vector, and then transition from one behavior to the other half-way through the episode by linearly interpolating the embeddings of the two demonstration trajectories over a window of 20 control steps. Although not always successful the learned controller often transitions robustly, despite not having been trained to do so. An example of this can be seen in the gif above. More examples can be seen in this [https://www.youtube.com/watch?v=VBrIll0B24o video].<br />
<br />
=Conclusions=<br />
The authors have proposed an approach for imitation learning that combines the favorable properties of techniques for density modeling with latent variables (VAEs) with those of GAIL. The result is a model that from a moderate number of demonstration tragectories, can learn:<br />
# a semantically well structured embedding of behaviors, <br />
# a corresponding multi-task controller that allows to robustly execute diverse behaviors from this embedding space, as well as <br />
# an encoder that can map new trajectories into the embedding space and hence allows for one-shot imitation. <br />
The experimental results demonstrate that this approach can work on a variety of control problems and that it scales even to very challenging ones such as the control of a simulated humanoid with a large number of degrees of freedoms.<br />
<br />
=Critique=<br />
The paper proposes a deep-learning-based approach to imitation learning which is sample-efficient and is able to imitate many diverse behaviors. The architecture can be seen as conditional generative adversarial imitation learning (GAIL). The conditioning vector is an embedding of a demonstrated trajectory, provided by a variational autoencoder. This results in one-shot imitation learning: at test time, a new demonstration can be embedded and provided as a conditioning vector to the imitation policy. The authors evaluate the method on several simulated motor control tasks.<br />
<br />
Pros:<br />
* Addresses a challenging problem of learning complex dynamics controllers / control policies<br />
* Well-written introduction / motivation<br />
* The proposed approach is able to learn complex and diverse behaviors and outperforms both the VAE alone (quantitatively) and GAIL alone (qualitatively).<br />
* Appealing qualitative results on the three evaluation problems. Interesting experiments with motion transitioning. <br />
<br />
Cons:<br />
* Comparisons to baselines could be more detailed.<br />
* Many key details are omitted (either on purpose, placed in the appendix, or simply absent, like the lack of definitions of terms in the modeling section, details of the planner model, simulation process, or the details of experimental settings)<br />
* Experimental evaluation is largely subjective (videos of robotic arm/biped/3D human motion)<br />
* A discussion of sample efficiency compared to GAIL and VAE would be interesting.<br />
* The presentation is not always clear, in particular, I had a hard time figuring out the notation in Section 3.<br />
* There has been some work on hybrids of VAEs and GANs, which seem worth mentioning when generative models are discussed, like:<br />
# Autoencoding beyond pixels using a learned similarity metric, Larsen et al., ICML 2016<br />
# Generating Images with Perceptual Similarity Metrics based on Deep Networks, Dosovitskiy&Brox. NIPS 2016<br />
These works share the intuition that good coverage of VAEs can be combined with sharp results generated by GANs.<br />
* Some more extensive analysis of the approach would be interesting. How sensitive is it to hyperparameters? How important is it to use VAE, not usual AE or supervised learning? How difficult will it be for others to apply it to new tasks?<br />
<br />
=References=<br />
# Duan, Y., Andrychowicz, M., Stadie, B., Ho, J., Schneider, J., Sutskever, I., Abbeel, P., & Zaremba, W. (2017). One-shot imitation learning. Preprint arXiv:1703.07326.<br />
# Ross, Stéphane, and Drew Bagnell. "Efficient reductions for imitation learning." Proceedings of the thirteenth international conference on artificial intelligence and statistics. 2010.<br />
# Wang, Z., Merel, J. S., Reed, S. E., de Freitas, N., Wayne, G., & Heess, N. (2017). Robust imitation of diverse behaviors. In Advances in Neural Information Processing Systems (pp. 5326-5335).<br />
# Producing flexible behaviours in simulated environments. (n.d.). Retrieved March 25, 2018, from https://deepmind.com/blog/producing-flexible-behaviours-simulated-environments/<br />
# Cmu humanoid. (2017, May 19). Retrieved March 25, 2018, from https://www.youtube.com/watch?v=NaohsyUxpxw<br />
# Cmu transitions. (2017, May 19). Retrieved March 25, 2018, from https://www.youtube.com/watch?v=VBrIll0B24o</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Understanding_Image_Motion_with_Group_Representations&diff=36042Understanding Image Motion with Group Representations2018-04-03T03:13:01Z<p>J3ku: Add commentary on KITTI results</p>
<hr />
<div>== Introduction ==<br />
Motion perception is a key component of computer vision. It is critical to problems such as optical flow and visual odometry, where a sequence of images are used to calculate either the pixel level (local) motion or the motion of the entire scene (global). The smooth image transformation caused by camera motion is a subspace of all position image transformations. Here, we are interested in realistic transformations caused by motion, therefore unrealistic motion caused by say, face swapping, is not considered. <br />
<br />
Supervised learning of 3D motion is challenging since explicit motion labels are not trivial to obtain. The proposed learning method does not need labeled data. Instead, the method applies constraints to learning by using the properties of motion space. The paper presents a general model of visual motion, and how the motion space properties of associativity and invertibility can be used to constrain the learning of a deep neural network. The results show evidence that the learned model captures motion in both 2D and 3D settings. This method can be used to extract useful information for vehicle localization, tracking, and odometry.<br />
<br />
[[File:paper13_fig1.png|650px|center|]]<br />
<br />
== Related Work ==<br />
The most common global representations of motion are from structure from motion (SfM) and simultaneous localization and mapping (SLAM), which represent poses in special Euclidean group <math> SE(3) </math> to represent a sequence of motions. However, these cannot be used to represent non-rigid or independent motions. The most used method for local representation is optical flow, which estimates motion by pixel over 2-D image. Furthermore, scene flow is a more generalized method of optical flow which estimates the point trajectories from 3-D motions. The limitation of optical flow is that it only captures motion locally, which makes capturing the overall motion impossible. <br />
<br />
Another approach to representing motion is spatiotemporal features (STFs), which are flexible enough to represent non-rigid motions since there is usually a dimensionality reduction process involved. However these approaches are restricted to fixed windows of representation. <br />
<br />
There are also works using CNN’s to learn optical flow using brightness constancy assumptions, and/or photometric local constraints. Works on stereo depth estimation using learning has also shown results. Regarding image sequences, there are works on shuffling the order of images to learn representations of its contents, as well as learning representations equivariant to the egomotion of the camera.<br />
<br />
== Approach ==<br />
The proposed method is based on the observation that 3D motions, equipped with composition, form a group. By learning the underlying mapping that captures the motion transformations, we are approximating latent motion of the scene. The method is designed to capture group associativity and invertibility.<br />
<br />
Consider a latent structure space <math>S</math>, element of the structure space generates images via projection <math>\pi:S\rightarrow I</math>, latent motion space <math>M</math> which is some closed subgroup of the set of homeomorphism on <math>S</math>. For <math>s \in S</math>, a continuous motion sequence <math> \{m_t \in M | t \geq 0\} </math> generates continous image sequence <math> \{i_t \in I | t \geq 0\} </math> where <math> i_t=\pi(m_t(s)) </math>. Writing this as a hidden Markov model gives <math> i_t=\pi(m_{\Delta t}(s_{t-1}))) </math> where the current state is based on the change from the previous. Since <math> M </math> is a closed group on <math> S </math>, it is associative, has inverse, and contains idenity. <math> SE(3) </math> is an exmaple of this. To be more specific, the latent structure of a scene from rigid image motion could be modelled by a point cloud with a motion space <math>M=SE(3)</math>, where rigid image motion can be produced by a camera translating and rotating through a rigid scene in 3D. When a scene has N rigid bodies, the motion space can be represented as <math>M=[SE(3)]^N</math>.<br />
<br />
=== Learning Motion by Group Properties ===<br />
The goal is to learn a function <math> \Phi : I \times I \rightarrow \overline{M} </math>, <math> \overline{M} </math> indicating mapping of image pairs from <math> M </math> to its representation, as well as the composition operator <math> \diamond : \overline{M} \rightarrow \overline{M} </math> that emulates the composition of these elements in <math> M </math>. For all sequences, it is assumed that for all times <math> t_0 < t_1 < t_2 ... </math>, the sequence representation should have the following properties: <br />
# Associativity: <math> \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_2}, I_{t_3}) = (\Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_2})) \diamond \Phi(I_{t_2}, I_{t_3}) = \Phi(I_{t_0}, I_{t_1}) \diamond (\Phi(I_{t_1}, I_{t_2}) \diamond \Phi(I_{t_2}, I_{t_3})) = \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_3})</math>, which means that the motion of differently composed subsequences of a sequence are equivalent<br />
# Has Identity: <math> \Phi(I_{t_0}, I_{t_1}) \diamond e = \Phi(I_{t_0}, I_{t_1}) = e \diamond \Phi(I_{t_0}, I_{t_1}) </math> and <math> e=\Phi(I_{t}, I_{t}) \forall t </math>, where <math>e</math> is the null image motion and the unique identity in the latent space<br />
# Invertibility: <math> \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_0}) = e </math>, so the inverse of the motion of an image sequence is the motion of that image sequence reversed<br />
An embedding loss is used to approximately enforce associativity and invertibility among subsequences sampled from image sequence. Associativity is encouraged by pushing sequences with the same final motion but different transitions to the same representation. Invertibility is encouraged by pushing sequences corresponding to the same motion with but in opposite directions away from each other, as well as pushing all loops to the same representation. Uniqueness of the identity is encouraged by pushing loops away from non-identity representations. Loops from different sequences are also pushed to the same representation (the identity).<br />
<br />
These constraints are true to any type of transformation resulting from image motion. This puts little restriction on the learning problems and allows all features relevant to the motion structure to be captured. On the other hand, optical flow assumes unchanging brightness between frames of the same projected scene, and motion estimates would degrade when that assumption does not hold.<br />
<br />
Also with this method, it is possible multiple representations <math> \overline{M} </math> can be learned from a single <math> M </math>, thus the learned representation is not necessary unique. In addition, the scenes are not expected to have rapid changing content, scene cuts, or long-term occlusions.<br />
<br />
=== Sequence Learning with Neural Networks ===<br />
The functions <math> \Phi </math> and <math> \diamond </math> are approximated by CNN and RNN, respectively. LSTM is used for RNN. The input to the network is a sequence of images <math> I_t = \{I_1,...,I_t\} </math>. The CNN processes pairs of images and generates intermediate representations, and the LSTM operates over the sequence of CNN outputs to produce an embedding sequence <math> R_t = \{R_{1,2},...,R_{t-1,t}\} </math>. Only the embedding at the final time step is used for loss. The network is trained to minimize a hinge loss with respect to embeddings to pairs of sequences as defined below:<br />
<br />
<center><math>L(R^1,R^2) = \begin{cases} d(R^1,R^2), & \text{if positive pair} \\ max(0, m - d(R^1,R^2)), & \text{if negative pair} \end{cases}</math></center><br />
<center><math> d_{cosine}(R^1,R^2)=1-\frac{\langle R^1,R^2 \rangle}{\lVert R^1 \rVert \lVert R^2 \rVert} </math></center><br />
<br />
where <math>d(R^1,R^2)</math> measure the distance between the embeddings of two sequences used for training selected to be cosine distance, <math> m </math> is a fixed scalar margin selected to be 0.5. Positive pairs are training examples where two sequences have the same final motion, negative pairs are training examples where two sequences have the exact opposite final motion. Using L2 distances yields similar results as cosine distances.<br />
<br />
Each training sequence is recomposed into 6 subsequences: two forward, two backward, and two identity. To prevent the network from only looking at static differences, subsequence pairs are sampled such that they have the same start and end frames but different motions in between. Sequences of varying lengths are also used to generalize motion on different temporal scales. Training the network with only one input images per time step was also tried, but consistently yielded work results than image pairs.<br />
<br />
[[File:paper13_fig2.png|650px|center|]]<br />
<br />
Overall, training with image pairs resulted in lower error than training with just single images. This is demonstrated in the below table.<br />
<br />
<br />
[[File:table.png|700px|center|]]<br />
<br />
== Experimentation ==<br />
Trained network using rotated and translated MNIST dataset as well as KITTI dataset. <br />
* Used Torch<br />
* Used Adam for optimization, decay schedule of 30 epochs, learning rate chosen by random serach<br />
* 50-60 batch size for MNIST, 25-30 batch size for KITTI<br />
* Dilated convolution with Relu and batch normalization<br />
* Two LSTM cell per layer 256 hidden units each<br />
* Sequence length of 3-5 images<br />
* MINIST networks with up to 12 images <br />
<br />
=== Rigid Motion in 2D ===<br />
* MNIST data rotated <math>[0, 360)</math> degrees and translated <math>[-10, 10] </math> pixels, i.e. <math>SE(2)</math> transformations<br />
* Visualized the representation using t-SNE<br />
** Clear clustering by translation and rotation but not object classes<br />
** Suggests the representation captures the motion properties in the dataset, but is independent of image contents<br />
* Visualized the image-conditioned saliency maps<br />
**In Figure 3, the red represents the positive gradients of the activation function with respect to the input image, and the negative gradients are represented in blue.<br />
**If we consider a saliency map as a first-order Taylor expansion, then the map could show the relationship between pixel and the representation.<br />
** Take derivative of the network output respect to the map<br />
** The area that has the highest gradient means that part contributes the most to the output<br />
** The resulting salient map strongly resembles spatiotemporal energy filters of classical motion processing<br />
** Suggests the network is learning the right motion structure<br />
<br />
[[File:paper13_fig3.png|700px|center|]]<br />
<br />
=== Real World Motion in 3D ===<br />
* Uses KITTI dataset collected on a car driving through roads in Germany<br />
* On a separate dataset with ground truth camera pose, linearly regress the representation to the ground truth<br />
** The result is compared against self supervised flow algorithm Yu et al.(2016) after the output from the flow algorithm is downsampled, then feed through PCA, then regressed against the camera motion<br />
** The data shows it performs not as well as the supervised algorithm, but consistently better than chance (guessing the mean value)<br />
** Largest improvements are shown in X and Z translation, which also have the most variance in the data<br />
** Shows the method is able to capture dominant motion structure<br />
* Test performance on interpolation task<br />
** Check <math>R([I_1,I_T])</math> against <math>R([I_1, I_m, I_T])</math>, <math>R([I_1, I_{IN}, I_T])</math>, and <math>R([I_1, I_{OUT}, I_T])</math><br />
** Test how sensitive the network is to deviations from unnatural motion<br />
** High errors <math>\gg 1</math> means the network can distinguish between realistic and unrealistic motion<br />
** In order to do this, the distance between the embeddings of the frame sequences of the first and last frame <math>R([I_1,I_T])</math> and of the first, middle, and last frame <math>R([I_1, I_m, I_T])</math> is computed. This distance is compared with the distance when the middle frame of the second embedding is changed to a frame that is visually similar (inside sequence): <math>R([I_1, I_{IN}, I_T])</math> and one that is visually dissimilar (outside sequence): <math>R([I_1, I_{OUT}, I_T])</math>. The results are shown in Table 3. The embedding distance method is compared to the Euclidean distance which is defined as the mean pixel distance between the test frame and <math>{I_1,I_T}</math>, whichever is smaller. It can be seen from the results that the embedding distance of the true frame is significantly lower than other frames. This means that the embedding distance used in the network is more sensitive to any atypical motions of the scenes. <br />
* Visualized saliency maps<br />
** Highlights objects moving in the background, and motion of the car in the foreground<br />
** Suggests the method can be used for tracking as well<br />
<br />
[[File:paper13_tab2.png|700px|center|]]<br />
<br />
[[File:paper13_fig4.png|700px|center|]]<br />
<br />
[[File:paper13_fig5.png|700px|center|]]<br />
<br />
[[File:table3_motion.PNG|700px|center|]]<br />
<br />
* Figure 7 displays graphs comparing the mean squared error of the method presented in this paper to the baseline chance method and the supervised Flow PCA method.<br />
<br />
[[File:paper13_fig6.PNG|700px|center|]]<br />
<br />
== Conclusion ==<br />
The authors presented a new model of motion and a method for learning motion representations. It is shown that by enforcing group properties we can learn motion representations that are able to generalize between scenes with disparate content. The results can be useful for navigation, prediction, and other behavioral tasks relying on motion. Due to the fact that this method does not require labelled data, it can be applied to a large variety of tasks.<br />
<br />
== Criticism ==<br />
Although this method does not require any labelled data, it is still learning by supervision through defined constraints. The idea of training using unlabelled data is interesting and it does have meaningful practical application. Unfortunately, the author did not provide convincing experimental results. Results from motion estimation problems are typically compared against ground truth data for their accuracy. The author performed experiments on transformed MNIST data and KITTI data. The MNIST data is transformed by the author, thus the ground truth is readily available. However the author only claimed the validity of the results through indirect means of using t-SNE and saliency map visualization. For the KITTI dataset, the author regressed the representations against ground truth for some mapping from the network output to some physical motion representation. Again, the results were compared only indirectly against ground truth, also shows poor results when compared with the Flow+PCA baseline, especially for X and Z translations as well as Y rotation, which are the main elements of motion present in the KITTI dataset. Such experimentation made the method hardly convincing and applicable to real-world applications. In addition, the network does not output motion representations with physical meanings, making the proposed method useless for any real world applications.<br />
<br />
One of the motivations the authors use for this approach is that traditional SLAM formulations represent motion as a sequence of poses in <math> SE(3) </math>, and that they are unable to represent non-rigid or independent motions. There exist SLAM formulations that represent motion as [http://ieeexplore.ieee.org/document/7353368/ Gaussian processes], as well as [http://journals.sagepub.com/doi/abs/10.1177/0278364915585860 temporal basis functions], and it is quite [https://openslam.org/robotvision.html common] for inertial, monocular-camera SLAM problems to use a motion representation on <math> SIM(3) </math>, which is the group containing all scale-preserving transformations. A <math> SIM(3) </math> transformation is not, in general, rigid, so it is not true to say that modern SLAM is unable to represent non-rigid motions. Additionally, the saliency images from the KITTI experiment displaying network gradients on independently moving objects in the scene does not necessarily mean that the motion representation is capturing independent motion, it just means that the network representation is dependent on those pixels. As the authors did not provide an error comparison between images containing independent motions and those without, it is possible that these network gradients only contribute to error (in terms of the camera pose) instead of capturing independent motions.<br />
<br />
Another criticism is that the group-properties constraint the authors impose is too weak. Any set consisting of functions, their inverses, and the identity forms a group. While physical motions are one example of such a group, there are many valid groups that do not represent any coherent physical motions.<br />
<br />
Since the network has to learn both the group elements, <math>\overline{M}</math>, and the composition function, <math>\diamond</math>, associated with the group it is difficult to tell how each of them are performing.<br />
<br />
Finally, the method requires domain knowledge of the motion space and feature engineering for encoding it, which reduces the ease with which the method can be generalized to various tasks.<br />
<br />
== References ==<br />
Jaegle, A. (2018). Understanding image motion with group representations . ICLR. Retrieved from https://openreview.net/pdf?id=SJLlmG-AZ.<br />
<br />
Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Deep inside convolutional networks: Visualising image classification models and saliency maps. In ''International Conference on Learning Representations (ICLR) Workshop'', 2013.<br />
<br />
Jason J Yu, Adam W Harley, and Konstantinos G Derpanis. Back to basics: Unsupervised learning of optical flow via brightness constancy and motion smoothness. ''European Conference on Computer Vision (ECCV) Workshops'', 2016.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Predicting_Floor-Level_for_911_Calls_with_Neural_Networks_and_Smartphone_Sensor_Data&diff=36041stat946w18/Predicting Floor-Level for 911 Calls with Neural Networks and Smartphone Sensor Data2018-04-03T02:57:52Z<p>J3ku: Add example showing importance in an emergency. Add link to code on GitHub</p>
<hr />
<div>= Introduction =<br />
During emergency 911 calls, knowing the exact position of the victims is crucial to a fast response and a successful rescue. Knowing the victim's floor level in an emergency can speed up the search by a factor proportional to the number of floors in the building. Problems arise when the caller is unable to give their physical position accurately. This can happen for instance when the caller is disoriented, held hostage, or a child is calling on behalf of the victim. GPS sensors on smartphones can provide the rescuers with the geographic location. However GPS fails to give an accurate floor level inside a tall building. Previous work have explored using Wi-Fi signals or beacons placed inside the buildings, but these methods are not self-contained and require prior infrastructure knowledge.<br />
<br />
Fortunately, today’s smartphones are equipped with many more sensors including barometers and magnetometers. Deep learning can be applied to predict floor level based on these sensor readings. <br />
Firstly, an LSTM is trained to classify whether the caller is indoors or outdoors using GPS, RSSI (Received Signal Strength Indication), and magnetometer sensor readings. Next, an unsupervised clustering algorithm is used to predict the floor level depending on the barometric pressure difference. With these two parts working together, a self-contained floor level prediction system can achieve 100% accuracy, without any external prior knowledge.<br />
<br />
This paper is published in ICLR 2018. The code, data, and app are open-source on [https://github.com/williamFalcon/Predicting-floor-level-for-911-Calls-with-Neural-Networks-and-Smartphone-Sensor-Data (GitHub)]<br />
<br />
= Data Description =<br />
The authors developed an iOS app called Sensory and used it to collect data on an iPhone 6. The following sensor readings were recorded: indoors, created at, session id, floor, RSSI strength, GPS latitude, GPS longitude, GPS vertical accuracy, GPS horizontal accuracy, GPS course, GPS speed, barometric relative altitude, barometric pressure, environment context, environment mean building floors, environment activity, city name, country name, magnet x, magnet y, magnet z, magnet total.<br />
<br />
The indoor-outdoor data has to be manually entered as soon as the user enters or exits a building. To gather the data for floor level prediction, the authors conducted 63 trials among five different buildings throughout New York City. The actual floor level was recorded manually for validation purposes only, since unsupervised learning is being used.<br />
<br />
= Methods =<br />
The proposed method first determines if the user is indoor or outdoor and detects the instances of transition between them. When an outdoor to indoor transition event occurs, the elevation of the user is saved using an estimation from the cellphone barometer. Finally, the exact floor level is predicted through clustering techniques. Indoor/outdoor classification is critical to the working of this method. Once the user is detected to be outdoors, he is assumed to be at the ground level. The vertical height and floor estimation is applied only when the user is indoors. The indoor/outdoor transitions are used to save the barometer readings at the ground level for use as reference pressure.<br />
<br />
=== Indoor/Outdoor Classification === <br />
<br />
An LSTM network is used to solve the indoor-outdoor classification problem. Here is a diagram of the network architecture.<br />
<br />
[[File:lstm.jpg | 500px]]<br />
<br />
Figure 1: LSTM network architecture. A 3-layer LSTM. Inputs are sensor readings for d consecutive time-steps. Target is y = 1 if indoors and y = 0 if outdoors.<br />
<br />
<math> X_i</math> contains a set of <math>d</math> consecutive sensor readings, i.e. <math> X_i = [x_1, x_2,...,x_d] </math>. <math>Y</math> is labelled as 0 for outdoors and 1 for indoors. <math>d</math> is chosen to be 3 by random-search so that <math>X</math> has 3 points <math>X_i = [x_{j-1}, x_j, x_{j+1}]</math> and the middle <math>x_j</math> is used for the <math>y</math> label.<br />
The LSTM contains three layers. Layers one and two have 50 neurons followed by a dropout layer set to 0.2. Layer 3 has two neurons fed directly into a one-neuron feedforward layer with a sigmoid activation function. The input is the sensor readings, and the output is the indoor-outdoor label. The objective function is the cross-entropy between the true labels and the predictions.<br />
<br />
\begin{equation}<br />
C(y_i, \hat{y}_i) = \frac{1}{n} \sum_{i=1}^{n} -(y_i log(\hat{y_i}) + (1 - y_i) log(1 - \hat{y_i}))<br />
\label{equation:binCE}<br />
\end{equation}<br />
<br />
The main reason why the neural network is able to predict whether the user is indoors or outdoors is that it learns a pattern of how the walls of buildings interfere with the GPS signals. The LSTM is able to find the pattern in the GPS signal strength in combination with other sensor readings to give an accurate prediction. However, the change in GPS signal does not happen instantaneously as the user walks indoor. Thus, a window of 20 seconds is allowed, and the minimum barometric pressure reading within that window is recorded as the ground floor.<br />
<br />
=== Indoor/Outdoor Transition === <br />
To determine the exact time the user makes an indoor/outdoor transition, two vector masks are convolved across the LSTM predictions.<br />
<br />
\begin{equation}<br />
V_1 = [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]<br />
\end{equation} <br />
<br />
\begin{equation}<br />
V_2 = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]<br />
\end{equation}<br />
<br />
The Jaccard distances measures the similarity of two sets and is calculated with the following equation:<br />
<br />
\begin{equation}<br />
J_j = J(s_i, V_j) = \frac{|s_i \cap V_j|}{|s_i| + |V_j| - |s_i \cap V_j|} <br />
\label{equation:Jaccard}<br />
\end{equation}<br />
<br />
If the Jaccard distance between <math>V_{1}</math> and sub-sequence <math> s_i </math> is greater or equal to the threshold 0.4, it means there was a transition from indoors to outdoors in the vicinity of the 20 second range of the vector mask. Similarly, a distance of to 0.4 or greater to <math>V_{2}</math> indicates a transition from outdoors to indoors. Sets of transition windows are merged together if they occur close in time to each other, with the average transition time of both windows being used as the new transition time.<br />
<br />
[[File:FindIOIndexes.png | 700px]]<br />
<br />
=== Vertical Height Estimation === <br />
Once the barometric pressure of the ground floor is known, the user’s current relative altitude can be calculated by the international pressure equation, where <math>m_\Delta</math> is the estimated height, <math> p_1 </math> is the pressure reading of the device, and <math> p_0 </math> is the reference pressure at ground level while transitioning from outdoor to indoor.<br />
<br />
\begin{equation}<br />
m_\Delta = f_{floor}(p_0, p_1) = 44330 (1 - (\frac{p_1}{p_0})^{\frac{1}{5.255}})<br />
\label{equation:baroHeight}<br />
\end{equation}<br />
<br />
In appendix B.1, the authors acknowledge that for this system to work, pressures variations due to weather or temperature must be accounted for as those variations are on the same order of magnitude or larger than the pressure variations caused by changing altitude. They suggest using a nearby reference station with known altitude to continuously measure and correct for this effect.<br />
<br />
=== Floor Estimation === <br />
Given the user’s relative altitude, the floor level can be determined. However, this is not a straightforward task because different buildings have different floor heights, different floor labeling (E.g. not including the 13th floor), and floor heights within the same building can vary from floor to floor. To solve these problems, altitude data collected are clustered into groups. Each cluster represents the approximate altitude of a floor.<br />
<br />
Here is an example of altitude data collected across 41 trials in the Uris Hall building in New York City. Each dashed line represent the center of a cluster.<br />
<br />
[[File:clusters.png | 500px]]<br />
<br />
Figure 2: Distribution of measurements across 41 trials in the Uris Hall building in New York City. A clear size difference is specially noticeable at the lobby. Each dotted line corresponds to an actual floor in the building learned from clustered data-points.<br />
<br />
Here is the algorithm for the floor level prediction.<br />
<br />
[[File:PredictFloor.png | 700px]]<br />
<br />
= Experiments and Results =<br />
The authors performed evaluation on two different tasks: The indoor-outdoor classification task and the floor level prediction task. In the indoor-outdoor detection task, they compared six different models, LSTM, feedforward neural networks, logistic regression, SVM, HMM and Random Forests. In the floor level prediction task, they evaluated the full system.<br />
<br />
== Indoor-Outdoor Classification Results ==<br />
Here are the results for the indoor-outdoor classification problem using different machine learning techniques. LSTM has the best performance on the test set.<br />
The LSTM is trained for 24 epochs with a batch size of 128. All the hyper-parameters such as learning rate(0.006), number of layers, d size, number of hidden units and dropout rate were searched through random search algorithm.<br />
<br />
[[File:IOResults.png]]<br />
<br />
== Floor Level Prediction Results ==<br />
The following are the results for the floor level prediction from the 63 collected samples. Results are given as the percent which matched the floor exactly, off by one, or off by more than one. In each column, the left number is the accuracy using a fixed floor height, and the number on the right is the accuracy when clustering was used to calculate a variable floor height. It was found that using the clustering technique produced 100% accuracy on floor predictions. The conclusion from these results is that using building-specific floor heights produces significantly better results.<br />
<br />
[[File:FloorLevelResults.png]]<br />
<br />
== Floor Level Clustering Results == <br />
Here is the comparison between the estimated floor height and the ground truth in the Uris Hall building.<br />
<br />
[[File:FloorComparison.png]]<br />
<br />
= Criticism =<br />
This paper is an interesting application of deep learning and achieves an outstanding result of 100% accuracy. However, it offers no new theoretical discoveries. The machine learning techniques used are fairly standard. The neural networks used in this paper only contains 3 layers, and the clustering is applied on one-dimensional data. This leads to the question whether deep learning is necessary and suitable for this task.<br />
<br />
It was explained in the paper that there are many cases where the system does not work. Some cases that were mentioned include: buildings with glass walls, delayed GPS signals, <br />
and pressure changes caused by air conditioning. Other examples I can think of are: uneven floors with some area higher than others, floors rarely visited, and tunnels from one building to another. These special cases are not specifically mentioned in the paper, but they do note that differences between outdoors and pressure-sealed buildings is a problem<br />
<br />
Another weakness of the method comes from the clustering technique. It requires a fair bit of training data. The author suggested two approaches. First, the data can be stored in the individual smartphone. This is not realistic as most people do not visit every single floor of every building, even if it is their own apartment buildings. The second approach is to let a central system (emergency department) collect data from multiple users (which is what the paper’s results are based on). However, such data collection would need to be done in accordance with local laws. Perhaps a better solution would be to use elevation reading to estimate a floor based on typical floor height. Even having a small range of floors of interest could help first responders significantly narrow down their response time.<br />
<br />
Aside from all the technical issues, if knowing the exact floor is required, would it maybe be easier to let the rescuers carry a barometer with them and search for the floor with the transmitted pressure reading?<br />
<br />
= Conclusion =<br />
This paper presented a novel deep learning application in predicting the floor level given sensory data from mobile phones. While there are no new theoretical discoveries, the application is novel and important for 911-responders; indeed, previous studies have shown that survival rates for urgent medical events drop exponentially for each floor increase. Although much of this is attributed to the actual floor height, this situation makes it all the more important to reduce ground-to-floor travel time.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=On_The_Convergence_Of_ADAM_And_Beyond&diff=36040On The Convergence Of ADAM And Beyond2018-04-03T02:50:45Z<p>J3ku: Add introduction to other optimizers</p>
<hr />
<div>= Introduction =<br />
Stochastic gradient descent (SGD) is currently the dominant method of training deep networks. Variants of SGD that scale the gradients using information from past gradients have been very successful, since the learning rate is adjusted on a per-feature basis, with ADAGRAD being one example. However, ADAGRAD performance deteriorates when loss functions are nonconvex and gradients are dense. Several variants of ADAGRAD, such as RMSProp, ADAM, ADADELTA, and NADAM have been proposed, which address the issue by using exponential moving averages of squared past gradients, thereby limiting the update to only rely on the past few gradients.<br />
<br />
This paper focuses strictly on the pitfalls in convergence of the ADAM optimizer from a theoretical standpoint and proposes a novel improvement to ADAM called AMSGrad. The paper introduces the idea that it is possible for ADAM to get "stuck" in its weighted average history, preventing it from converging to an optimal solution. For example, in an experiment there may be a large spike in the gradient during some minibatches, but since ADAM weighs the current update by the exponential moving averages of squared past gradients, the effect of the large spike in gradient is lost. To tackle these issues, several variants of ADAGRAD The authors' analysis suggest that this can be prevented through novel but simple adjustments to the ADAM optimization algorithm, which can improve convergence. This paper is published in ICLR 2018.<br />
<br />
== Notation ==<br />
The paper presents the following framework as a generalization to all training algorithms, allowing us to fully define any specific variant such as AMSGrad or SGD entirely within it:<br />
<br />
[[File:training_algo_framework.png|700px|center]]<br />
<br />
Where we have <math> x_t </math> as our network parameters defined within a vector space <math> \mathcal{F} </math>. <math> \prod_{\mathcal{F}} (y) = </math> the projection of <math> y </math> on to the set <math> \mathcal{F} </math>.<br />
<math> \psi_t </math> and <math> \phi_t </math> correspond to arbitrary functions we will provide later, The former maps from the history of gradients to <math> \mathbb{R}^d </math> and the latter maps from the history of the gradients to positive semi definite matrices. And finally <math> f_t </math> is our loss function at some time <math> t </math>, the rest should be pretty self explanatory. Using this framework and defining different <math> \psi_t </math> , <math> \phi_t </math> will allow us to recover all different kinds of training algorithms under this one roof.<br />
<br />
=== SGD As An Example ===<br />
To recover SGD using this framework we simply select <math> \phi_t (g_1, \dotsc, g_t) = g_t</math>, <math> \psi_t (g_1, \dotsc, g_t) = I </math> and <math>\alpha_t = \alpha / \sqrt{t}</math>. It is easy to see that no transformations are ultimately applied to any of the parameters based on any gradient history other than the most recent from <math> \phi_t </math> and that <math> \psi_t </math> in no way transforms any of the parameters by any specific amount as <math> V_t = I </math> has no impact later on.<br />
<br />
=== ADAGRAD As Another Example ===<br />
<br />
To recover ADAGRAD, we select <math> \phi_t (g_1, \dotsc, g_t) = g_t</math>, <math> \psi_t (g_1, \dotsc, g_t) = \frac{\sum_{i=1}^{t} g_i^2}{t} </math>, and <math>\alpha_t = \alpha / \sqrt{t}</math>. Therefore, compared to SGD, ADAGRAD uses a different step size for each parameter, based on the past gradients for that parameter; the learning rate becomes <math> \alpha_t = \alpha / \sqrt{\sum_i g_{i,j}^2} </math> for each parameter <math> j </math>. The authors note that this scheme is quite efficient when the gradients are sparse.<br />
<br />
=== ADAM As Another Example ===<br />
Once you can convince yourself that the recovery of SGD from the generalized framework is correct, you should understand the framework enough to see why the following setup for ADAM will allow us to recover the behaviour we want. ADAM has the ability to define a "learning rate" for every parameter based on how much that parameter moves over time (a.k.a its momentum) supposedly to help with the learning process.<br />
<br />
In order to do this, we will choose <math> \phi_t (g_1, \dotsc, g_t) = (1 - \beta_1) \sum_{i=0}^{t} {\beta_1}^{t - i} g_t </math>, psi to be <math> \psi_t (g_1, \dotsc, g_t) = (1 - \beta_2)</math>diag<math>( \sum_{i=0}^{t} {\beta_2}^{t - i} {g_t}^2) </math>, and keep <math>\alpha_t = \alpha / \sqrt{t}</math>. This setup is equivalent to choosing a learning rate decay of <math>\alpha / \sqrt{\sum_i g_{i,j}}</math> for <math>j \in [d]</math>.<br />
<br />
From this, we can now see that <math>m_t </math> gets filled up with the exponentially weighted average of the history of our gradients that we have come across so far in the algorithm. And that as we proceed to update we scale each one of our parameters by dividing out <math> V_t </math> (in the case of diagonal it is just one over the diagonal entry) which contains the exponentially weighted average of each parameter's momentum (<math> {g_t}^2 </math>) across our training so far in the algorithm. Thus each parameter has its own unique scaling by its second moment or momentum. Intuitively, from a physical perspective, if each parameter is a ball rolling around in the optimization landscape what we are now doing is instead of having the ball change positions on the landscape at a fixed velocity (i.e. momentum of 0) the ball now has the ability to accelerate and speed up or slow down if it is on a steep hill or flat trough in the landscape (i.e. a momentum that can change with time).<br />
<br />
= <math> \Gamma_t </math>, an Interesting Quantity =<br />
Now that we have an idea of what ADAM looks like in this framework, let us now investigate the following:<br />
<br />
<center><math> \Gamma_{t + 1} = \frac{\sqrt{V_{t+1}}}{\alpha_{t+1}} - \frac{\sqrt{V_t}}{\alpha_t} </math></center><br />
<br />
Which essentially measure the change of the "Inverse of the learning rate" across time (since we are using alpha's as step sizes). A key observation is that for SGD and ADAGRAD, <math>\Gamma_t \succeq 0</math> for all <math>t \in [T]</math>, which simply follows from the update rules of SGD and ADAGRAD. Looking back to our example of SGD it's not hard to see that this quantity is strictly positive semidefinite, which leads to "non-increasing" learning rates, which is a desired property. However, that is not the case with ADAM, and can pose a problem in a theoretical and applied setting. The problem ADAM can face is that <math> \Gamma_t </math> can potentially be indefinite for <math>t \in [T]</math>, which the original proof assumed it could not be. The math for this proof is VERY long so instead we will opt for an example to showcase why this could be an issue.<br />
<br />
Consider the loss function <math> f_t(x) = \begin{cases} <br />
Cx & \text{for } t \text{ mod 3} = 1 \\<br />
-x & \text{otherwise}<br />
\end{cases} </math><br />
<br />
Where we have <math> C > 2 </math> and <math> \mathcal{F} </math> is <math> [-1,1] </math>. Additionally we choose <math> \beta_1 = 0 </math> and <math> \beta_2 = 1/(1+C^2) </math>. We then proceed to plug this into our framework from before. This function is periodic and it's easy to see that it has the gradient of C once and then a gradient of -1 twice every period. It has an optimal solution of <math> x = -1 </math> (from a regret standpoint), but using ADAM we would eventually converge at <math> x = 1 </math>, since <math> \psi_t </math> would scale down the <math> C </math> by a factor of almost <math> C </math> so that it's unable to "overpower" the multiple -1's.<br />
<br />
We formalize this intuition in the results below.<br />
<br />
'''Theorem 1.''' There is an online convex optimization problem where ADAM has non-zero average regret. i.e. <math>R_T/T\nrightarrow 0 </math> as <math>T\rightarrow \infty</math>.<br />
<br />
One might think that adding a small constant in the denominator of the update function can help avoid this issue by modifying the update for ADAM as follow:<br />
\begin{align}<br />
\hat x_{t+1} = x_t - \alpha_t m_t/\sqrt{V_t + \epsilon \mathbb{I}}<br />
\end{align}<br />
<br />
The selection of <math>\epsilon</math> appears to be crucial for the performance of the algorithm in practice. However, this work shows that for any constant <math>\epsilon > 0</math>, there exists an online optimization setting where ADAM has non-zero average regret asymptotically.<br />
<br />
'''Theorem 2.''' For any constant <math>\beta_1,\beta_2 \in [0,1)</math> such that <math>\beta_2 < \sqrt{\beta_2}</math>, there is an online convex optimization problem where ADAM has non-zero average regret i.e. <math>R_T/T\nrightarrow 0 </math> as <math>T\rightarrow \infty</math>.<br />
<br />
'''Theorem 3.''' For any constant <math>\beta_1,\beta_2 \in [0,1)</math> such that <math>\beta_2 < \sqrt{\beta_2}</math>, there is a stochastic convex optimization problem for which ADAM does not converge to the optimal solution.<br />
<br />
= AMSGrad as an improvement to ADAM =<br />
There is a very simple intuitive fix to ADAM to handle this problem. We simply scale our historical weighted average by the maximum we have seen so far to avoid the negative sign problem. There is a very simple one-liner adaptation of ADAM to get to AMSGRAD:<br />
[[File:AMSGrad_algo.png|700px|center]]<br />
<br />
Below are some simple plots comparing ADAM and AMSGrad, the first are from the paper and the second are from another individual who attempted to recreate the experiments. The two plots somewhat disagree with one another so take this heuristic improvement with a grain of salt.<br />
<br />
[[File:AMSGrad_vs_adam.png|900px|center]]<br />
<br />
Here is another example of a one-dimensional convex optimization problem where ADAM fails to converge<br />
<br />
[[File:AMSGrad_vs_adam3.png|900px|center]]<br />
<br />
[[File:AMSGrad_vs_adam2.png|700px|center]]<br />
<br />
= Conclusion =<br />
We have introduced a framework for which we could view several different training algorithms. From there we used it to recover SGD as well as ADAM. In our recovery of ADAM we investigated the change of the inverse of the learning rate over time to discover in certain cases there were convergence issues. We proposed a new heuristic AMSGrad to help deal with this problem and presented some empirical results that show it may have helped ADAM slightly. Thanks for your time.<br />
<br />
== Critique ==<br />
The contrived example which serves as the intuition to illustrate the failure of ADAM is not convincing, since we can construct similar failure examples for SGD as well. <br />
Consider the loss function <br />
<br />
<math> f_t(x) = \begin{cases} <br />
-x & \text{for } t \text{ mod 2} = 1 \\<br />
-\frac{1}{2} x^2 & \text{otherwise}<br />
\end{cases} <br />
</math><br />
<br />
where <math> x \in \mathcal{F} = [-a, 1], a \in [1, \sqrt{2}) </math>. The optimal solution is <math>x=1</math>, but starting from initial point <math>x_{t=0} \le -1</math>, SGD will converge to <math>x = -a</math><br />
<br />
= Source =<br />
1. Sashank J. Reddi and Satyen Kale and Sanjiv Kumar. "On the Convergence of Adam and Beyond." International Conference on Learning Representations. 2018</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=36039Word translation without parallel data2018-04-03T02:20:43Z<p>J3ku: Add examples of annotated data</p>
<hr />
<div>[[File:Toy_example.png]]<br />
<br />
= Presented by =<br />
<br />
Xia Fan<br />
<br />
= Introduction =<br />
<br />
Many successful methods for learning relationships between languages stem from the hypothesis that there is a relationship between the context of words and their meanings. This means that if an adequate representation of a language is found in a high dimensional space (this is called an embedding), then words similar to a given word are close to one another in this space (ex. some norm can be minimized to find a word with similar context). Historically, another significant hypothesis is that these embedding spaces show similar structures over different languages. That is to say that given an embedding space for English and one for Spanish, a mapping could be found that aligns the two spaces and such a mapping could be used as a tool for translation. Many papers exploit these hypotheses, but use large parallel datasets for training. Recently, to remove the need for supervised training, methods have been implemented that utilize identical character strings (ex. letters or digits) in order to try to align the embeddings. The downside of this approach is that the two languages need to be similar to begin with as they need to have some shared basic building block. The method proposed in this paper uses an adversarial method to find this mapping between the embedding spaces of two languages without the use of large parallel datasets.<br />
<br />
This paper introduces a model that either is on par, or outperforms supervised state-of-the-art methods, without employing any cross-lingual annotated data such as bilingual dictionaries or parallel corpora (large and structured sets of texts). This method uses an idea similar to GANs: it leverages adversarial training to learn a linear mapping from a source to distinguish between the mapped source embeddings and the target embeddings, while the mapping is jointly trained to fool the discriminator. Second, this paper extracts a synthetic dictionary from the resulting shared embedding space and fine-tunes the mapping with the closed-form Procrustes solution from Schonemann (1966). Third, this paper also introduces an unsupervised selection metric that is highly correlated with the mapping quality and that the authors use both as a stopping criterion and to select the best hyper-parameters. This paper is published in ICLR 2018.<br />
<br />
= Model =<br />
<br />
<br />
=== Estimation of Word Representations in Vector Space ===<br />
<br />
This model focuses on learning a mapping between the two sets such that translations are close in the shared space. Before talking about the model it used, a model which can exploit the similarities of monolingual embedding spaces should be introduced. Mikolov et al.(2013) use a known dictionary of n=5000 pairs of words <math> \{x_i,y_i\}_{i\in{1,n}} </math>. and learn a linear mapping W between the source and the target space such that <br />
<br />
\begin{align}<br />
W^*=argmin_{W{\in}M_d(R)}||WX-Y||_F \hspace{1cm} (1)<br />
\end{align}<br />
<br />
where d is the dimension of the embeddings, <math> M_d(R) </math> is the space of d*d matrices of real numbers, and X and Y are two aligned matrices of size d*n containing the embeddings of the words in the parallel vocabulary. <br />
<br />
Xing et al. (2015) showed that these results are improved by enforcing orthogonality constraint on W. In that case, equation (1) boils down to the Procrustes problem, a matrix approximation problem for which the goal is to find an orthogonal matrix that best maps two given matrices on the measure of the Frobenius norm. It advantageously offers a closed form solution obtained from the singular value decomposition (SVD) of <math> YX^T </math> :<br />
<br />
\begin{align}<br />
W^*=argmin_{W{\in}M_d(R)}||WX-Y||_F=UV^T\textrm{, with }U\Sigma V^T=SVD(YX^T).<br />
\end{align}<br />
<br />
<br />
This can be proven as follows. First note that <br />
\begin{align}<br />
&||WX-Y||_F\\<br />
&= \langle WX-Y, WX-Y\rangle_F\\ <br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
& =\text{Tr}(W^T \sum_i \sigma_i u_i v_i^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\text{ invariance of trace under cyclic permutations}\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\text{ Cauchy-Swarz inequality}\\<br />
&= \sum_i \sigma_i<br />
\end{align}<br />
where we have used the invariance of trace under cyclic permutations, Cauchy-Schwarz, and the orthogonality of the columns of U and V. Note that choosing <br />
\begin{align}<br />
W=UV^T<br />
\end{align}<br />
achieves the bound. This completes the proof.<br />
<br />
=== Domain-adversarial setting ===<br />
<br />
This paper shows how to learn this mapping W without cross-lingual supervision. An illustration of the approach is given in Fig. 1. First, this model learn an initial proxy of W by using an adversarial criterion. Then, it use the words that match the best as anchor points for Procrustes. Finally, it improve performance over less frequent words by changing the metric of the space, which leads to spread more of those points in dense region. <br />
<br />
[[File:Toy_example.png |frame|none|alt=Alt text|Figure 1: Toy illustration of the method. (A) There are two distributions of word embeddings, English words in red denoted by X and Italian words in blue denoted by Y , which we want to align/translate. Each dot represents a word in that space. The size of the dot is proportional to the frequency of the words in the training corpus of that language. (B) Using adversarial learning, we learn a rotation matrix W which roughly aligns the two distributions. The green stars are randomly selected words that are fed to the discriminator to determine whether the two word embeddings come from the same distribution. (C) The mapping W is further refined via Procrustes. This method uses frequent words aligned by the previous step as anchor points, and minimizes an energy function that corresponds to a spring system between anchor points. The refined mapping is then used to map all words in the dictionary. (D) Finally, we translate by using the mapping W and a distance metric, dubbed CSLS, that expands the space where there is high density of points (like the area around the word “cat”), so that “hubs” (like the word “cat”) become less close to other word vectors than they would otherwise (compare to the same region in panel (A)).]]<br />
<br />
Let <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> be two sets of n and m word embeddings coming from a source and a target language respectively. A model is trained is trained to discriminate between elements randomly sampled from <math> WX={Wx_1,...,Wx_n} </math> and Y, We call this model the discriminator. W is trained to prevent the discriminator from making accurate predictions. As a result, this is a two-player game, where the discriminator aims at maximizing its ability to identify the origin of an embedding, and W aims at preventing the discriminator from doing so by making WX and Y as similar as possible. This approach is in line with the work of Ganin et al.(2016), who proposed to learn latent representations invariant to the input domain, where in this case, a domain is represented by a language(source or target).<br />
<br />
1. Discriminator objective<br />
<br />
Refer to the discriminator parameters as <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> that a vector z is the mapping of a source embedding (as opposed to a target embedding) according to the discriminator. The discriminator loss can be written as:<br />
<br />
\begin{align}<br />
L_D(\theta_D|W)=-\frac{1}{n} \sum_{i=1}^n log P_{\theta_D}(source=1|Wx_i)-\frac{1}{m} \sum_{i=1}^m log P_{\theta_D}(source=0|y_i)<br />
\end{align}<br />
<br />
2. Mapping objective <br />
<br />
In the unsupervised setting, W is now trained so that the discriminator is unable to accurately predict the embedding origins: <br />
<br />
\begin{align}<br />
L_W(W|\theta_D)=-\frac{1}{n} \sum_{i=1}^n log P_{\theta_D}(source=0|Wx_i)-\frac{1}{m} \sum_{i=1}^m log P_{\theta_D}(source=1|y_i)<br />
\end{align}<br />
<br />
3. Learning algorithm <br />
To train the model, the authors follow the standard training procedure of deep adversarial networks of Goodfellow et al. (2014). For every input sample, the discriminator and the mapping matrix W are trained successively with stochastic gradient updates to respectively minimize <math> L_D </math> and <math> L_W </math><br />
<br />
=== Refinement procedure ===<br />
<br />
The matrix W obtained with adversarial training gives good performance (see Table 1), but the results are still not on par with the supervised approach. In fact, the adversarial approach tries to align all words irrespective of their frequencies. However, rare words have embeddings that are less updated and are more likely to appear in different contexts in each corpus, which makes them harder to align. Under the assumption that the mapping is linear, it is then better to infer the global mapping using only the most frequent words as anchors. Besides, the accuracy on the most frequent word pairs is high after adversarial training.<br />
To refine the mapping, this paper build a synthetic parallel vocabulary using the W just learned with adversarial training. Specifically, this paper consider the most frequent words and retain only mutual nearest neighbors to ensure a high-quality dictionary. Subsequently, this paper apply the Procrustes solution in (2) on this generated dictionary. Considering the improved solution generated with the Procrustes algorithm, it is possible to generate a more accurate dictionary and apply this method iteratively, similarly to Artetxe et al. (2017). However, given that the synthetic dictionary obtained using adversarial training is already strong, this paper only observe small improvements when doing more than one iteration, i.e., the improvements on the word translation task are usually below 1%.<br />
<br />
=== Cross-Domain Similarity Local Scaling (CSLS) ===<br />
<br />
This paper considers a bi-partite neighborhood graph, in which each word of a given dictionary is connected to its K nearest neighbors in the other language. <math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> to its target neighborhood as<br />
<br />
\begin{align}<br />
r_T(Wx_s)=\frac{1}{K}\sum_{y\in N_T(Wx_s)}cos(Wx_s,y_t)<br />
\end{align}<br />
<br />
where cos(.,.) is the cosine similarity. Likewise, the mean similarity of a target word <math> y_t </math> to its neighborhood is denoted as <math> r_S(y_t) </math>. This is used to define similarity measure CSLS(.,.) between mapped source words and target words as <br />
<br />
\begin{align}<br />
CSLS(Wx_s,y_t)=2cos(Wx_s,y_t)-r_T(Wx_s)-r_S(y_t)<br />
\end{align}<br />
<br />
This process increases the similarity associated with isolated word vectors, but decreases the similarity of vectors lying in dense areas. <br />
<br />
CSLS represents an improved measure for producing reliable matching words between two languages (i.e. neighbors of a word in one language should ideally correspond to the same words in the second language). The nearest neighbors algorithm is asymmetric, and in high-dimensional spaces, it suffers from the problem of hubness, in which some points are nearest neighbors to exceptionally many points, while others are not nearest neighbors to any points. Existing approaches for combating the effect of hubness on word translation retrieval involve performing similarity updates one language at a time without consideration for the other language in the pair (Dinu et al., 2015, Smith et al., 2017). Consequently, they yielded less accurate results when compared to CSLS in experiments conducted in this paper (Table 1).<br />
<br />
= Training and architectural choices =<br />
=== Architecture ===<br />
<br />
This paper use unsupervised word vectors that were trained using fastText2. These correspond to monolingual embeddings of dimension 300 trained on Wikipedia corpora; therefore, the mapping W has size 300 × 300. Words are lower-cased, and those that appear less than 5 times are discarded for training. As a post-processing step, only the first 200k most frequent words were selected in the experiments.<br />
For the discriminator, it use a multilayer perceptron with two hidden layers of size 2048, and Leaky-ReLU activation functions. The input to the discriminator is corrupted with dropout noise with a rate of 0.1. As suggested by Goodfellow (2016), a smoothing coefficient s = 0.2 is included in the discriminator predictions. This paper use stochastic gradient descent with a batch size of 32, a learning rate of 0.1 and a decay of 0.95 both for the discriminator and W . <br />
<br />
=== Discriminator inputs ===<br />
The embedding quality of rare words is generally not as good as the one of frequent words (Luong et al., 2013), and it is observed that feeding the discriminator with rare words had a small, but not negligible negative impact. As a result, this paper only feed the discriminator with the 50,000 most frequent words. At each training step, the word embeddings given to the discriminator are sampled uniformly. Sampling them according to the word frequency did not have any noticeable impact on the results.<br />
<br />
=== Orthogonality===<br />
In this work, the authors propose to use a simple update step to ensure that the matrix W stays close to an orthogonal matrix during training (Cisse et al. (2017)). Specifically, the following update rule on the matrix W is used :<br />
<br />
\begin{align}<br />
W \leftarrow (1+\beta)W-\beta(WW^T)W<br />
\end{align}<br />
<br />
where β = 0.01 is usually found to perform well. This method ensures that the matrix stays close to the manifold of orthogonal matrices after each update.<br />
<br />
This update rule can be justified as follows. Consider the function <br />
\begin{align}<br />
g: \mathbb{R}^{d\times d} \to \mathbb{R}^{d \times d}<br />
\end{align}<br />
defined by<br />
\begin{align}<br />
g(W)= W^T W -I.<br />
\end{align}<br />
<br />
The derivative of g at W is is the linear map<br />
\begin{align}<br />
Dg[W]: \mathbb{R}^{d \times d} \to \mathbb{R}^{d \times d}<br />
\end{align}<br />
defined by<br />
\begin{align}<br />
Dg[W](H)= H^T W + W^T H.<br />
\end{align}<br />
<br />
The adjoint of this linear map is<br />
<br />
\begin{align}<br />
D^\ast g[W](H)= WH^T +WH.<br />
\end{align}<br />
<br />
Now consider the function f<br />
\begin{align}<br />
f: \mathbb{R}^{d \times d} \to \mathbb{R}<br />
\end{align}<br />
<br />
defined by<br />
<br />
\begin{align}<br />
f(W)=||g(W) ||_F^2=||W^TW -I ||_F^2.<br />
\end{align}<br />
<br />
f has gradient:<br />
\begin{align}<br />
\nabla f (W) = 2D^\ast g[W] (g(W ) ) =2W(W^TW-I) +2W(W^TW-I)=4W W^TW-4W.<br />
\end{align}<br />
or<br />
\begin{align}<br />
\nabla f (W) = \nabla||W^TW-I||_F = \nabla\text{Tr}(W^TW-I)(W^TW-I)=4(\nabla(W^TW-I))(W^TW-I)=4W(W^TW-I)\text{ (check derivative of trace function)}<br />
\end{align}<br />
<br />
Thus the update<br />
\begin{align}<br />
W \leftarrow (1+\beta)W-\beta(WW^T)W<br />
\end{align}<br />
amounts to a step in the direction opposite the gradient of f. That is, a step toward the set of orthogonal matrices.<br />
<br />
=== Dictionary generation ===<br />
The refinement step requires the generation of a new dictionary at each iteration. In order for the Procrustes solution to work well, it is best to apply it on correct word pairs. As a result, the CSLS method is used to select more accurate translation pairs in the dictionary. To further increase the quality of the dictionary, and ensure that W is learned from correct translation pairs, only mutual nearest neighbors were considered, i.e. pairs of words that are mutually nearest neighbors of each other according to CSLS. This significantly decreases the size of the generated dictionary, but improves its accuracy, as well as the overall performance.<br />
<br />
=== Validation criterion for unsupervised model selection ===<br />
<br />
This paper consider the 10k most frequent source words, and use CSLS to generate a translation for each of them, then compute the average cosine similarity between these deemed translations, and use this average as a validation metric. Figure 2 below shows the correlation between the evaluation score and this unsupervised criterion (without stabilization by learning rate shrinkage)<br />
<br />
<br />
<br />
[[File:fig2_fan.png |frame|none|alt=Alt text|Figure 2: Unsupervised model selection.<br />
Correlation between the unsupervised validation criterion (black line) and actual word translation accuracy (blue line). In this particular experiment, the selected model is at epoch 10. Observe how the criterion is well correlated with translation accuracy.]]<br />
<br />
= Results =<br />
<br />
The results on word translation retrieval using the bilingual dictionaries are presented in Table 1, and a comparison to previous work in shown in Table 2 where the unsupervised model significantly outperforms previous approaches. The results on the sentence translation retrieval task are presented in Table 3, and the cross-lingual word similarity task in Table 4. Finally, the results on word-by-word translation for English-Esperanto are presented in Table 5. The bilingual dictionary used here does not account for words with multiple meanings.<br />
<br />
[[File:table1_fan.png |frame|none|alt=Alt text|Table 1: Word translation retrieval P@1 for the released vocabularies in various language pairs. The authors consider 1,500 source test queries, and 200k target words for each language pair. The authors use fastText embeddings trained on Wikipedia. NN: nearest neighbors. ISF: inverted softmax. (’en’ is English, ’fr’ is French, ’de’ is German, ’ru’ is Russian, ’zh’ is classical Chinese and ’eo’ is Esperanto)]]<br />
<br />
<br />
[[File:table2_fan.png |frame|none|alt=Alt text|English-Italian word translation average precisions (@1, @5, @10) from 1.5k source word queries using 200k target words. Results marked with the symbol † are from Smith et al. (2017). Wiki means the embeddings were trained on Wikipedia using fastText. Note that the method used by Artetxe et al. (2017) does not use the same supervision as other supervised methods, as they only use numbers in their ini- tial parallel dictionary.]]<br />
<br />
[[File:table3_fan.png |frame|none|alt=Alt text|Table 3: English-Italian sentence translation retrieval. The authors report the average P@k from 2,000 source queries using 200,000 target sentences. The authors use the same embeddings as in Smith et al. (2017). Their results are marked with the symbol †.]]<br />
<br />
[[File:table4_fan.png |frame|none|alt=Alt text|Table 4: Cross-lingual wordsim task. NASARI<br />
(Camacho-Collados et al. (2016)) refers to the official SemEval2017 baseline. The authors report Pearson correlation.]]<br />
<br />
[[File:table5_fan.png |frame|none|alt=Alt text|Table 5: BLEU score on English-Esperanto.<br />
Although being a naive approach, word-by- word translation is enough to get a rough idea of the input sentence. The quality of the gener- ated dictionary has a significant impact on the BLEU score.]]<br />
<br />
[[File:paper9_fig3.png |frame|none|alt=Alt text|Figure 3: The paper also investigated the impact of monolingual embeddings. It was found that model from this paper can align embeddings obtained through different methods, but not embeddings obtained from different corpora, which explains the large performance increase in Table 2 due to the corpus change from WaCky to Wiki using CBOW embedding. This is conveyed in this figure which displays English to English world alignment accuracies with regard to word frequency. Perfect alignment is achieved using the same model and corpora (a). Also good alignment using different model and corpora, although CSLS consistently has better results (b). Worse results due to use of different corpora (c). Even worse results when both embedding model and corpora are different.]]<br />
<br />
= Conclusion =<br />
It is clear that one major downfall of this method when it actually comes to translation is the restriction that the two languages must have similar intrinsic structures to allow for the embeddings to align. However, given this assumption, this paper shows for the first time that one can align word embedding spaces without any cross-lingual supervision, i.e., solely based on unaligned datasets of each language, while reaching or outperforming the quality of previous supervised approaches in several cases. Using adversarial training, the model is able to initialize a linear mapping between a source and a target space, which is also used to produce a synthetic parallel dictionary. It is then possible to apply the same techniques proposed for supervised techniques, namely a Procrustean optimization.<br />
<br />
= Open source code =<br />
The source code for the paper is provided at the following github link: https://github.com/facebookresearch/MUSE. The repository provides the source code as written in PyTorch by the authors of this paper.<br />
<br />
= Source =<br />
Dinu, Georgiana; Lazaridou, Angeliki; Baroni, Marco<br />
| Improving zero-shot learning by mitigating the hubness problem<br />
| arXiv:1412.6568<br />
<br />
Lample, Guillaume; Denoyer, Ludovic; Ranzato, Marc'Aurelio <br />
| Unsupervised Machine Translation Using Monolingual Corpora Only<br />
| arXiv: 1701.04087<br />
<br />
Smith, Samuel L; Turban, David HP; Hamblin, Steven; Hammerla, Nils Y<br />
| Offline bilingual word vectors, orthogonal transformations and the inverted softmax<br />
| arXiv:1702.03859<br />
<br />
Lample, G. (n.d.). Facebookresearch/MUSE. Retrieved March 25, 2018, from https://github.com/facebookresearch/MUSE</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Rethinking_the_Smaller-Norm-Less-Informative_Assumption_in_Channel_Pruning_of_Convolutional_Layers&diff=36034stat946w18/Rethinking the Smaller-Norm-Less-Informative Assumption in Channel Pruning of Convolutional Layers2018-04-03T02:05:20Z<p>J3ku: Clarify FLOPs and FLOPS, add example</p>
<hr />
<div>== Introduction ==<br />
<br />
With the recent and ongoing surge in low-power, intelligent agents (such as wearables, smartphones, and IoT devices), there exists a growing need for machine learning models to work well in resource-constrained environments. Deep learning models have achieved state-of-the-art on a broad range of tasks; however, they are difficult to deploy in their original forms. For example, AlexNet (Krizhevsky et al., 2012), a model for image classification, contains 61 million parameters and requires 1.5 billion floating point operations (FLOPs) in one inference pass. A more accurate model, ResNet-50 (He et al., 2016), has 25 million parameters but requires 4.08 FLOPs. A high-end desktop GPU such as a Titan Xp is capable of [https://www.nvidia.com/en-us/titan/titan-xp/ (12 TFLOPS (tera-FLOPs per second))], while the Adreno 540 GPU used in a Samsung Galaxy S8 is only capable of [https://gflops.surge.sh (567 GFLOPS)] which is less than 5% of the Titan Xp. Clearly, it would be difficult to deploy and run these models on low-power devices.<br />
<br />
In general, model compression can be accomplished using four main, not mutually exclusive methods (Cheng et al., 2017): weight pruning, quantization, matrix transformations, and weight tying. By not mutually exclusive, we mean that these methods can be used not only separately but also in combination for compressing a single model; the use of one method does not exclude any of the other methods from being viable. <br />
<br />
Ye et al. (2018) explores pruning entire channels in a convolutional neural network (CNN). Past work has mostly focused on norm[based or error-based heuristics to prune channels; instead, Ye et al. (2018) show that their approach is easily reproducible and has favorable qualities from an optimization standpoint. In other words, they argue that the norm-based assumption is not as informative or theoretically justified as their approach, and provide strong empirical evidence of these findings.<br />
<br />
== Motivation ==<br />
<br />
Some previous works on pruning channel filters (Li et al., 2016; Molchanov et al., 2016) have focused on using the L1 norm to determine the importance of a channel. Ye et al. (2018) show that, in the deep linear convolution case, penalizing the per-layer norm is coarse-grained; they argue that one cannot assign different coefficients to L1 penalties associated with different layers without risking the loss function being susceptible to trivial re-parameterizations. As an example, consider the following deep linear convolutional neural network with modified LASSO loss:<br />
<br />
$$\min \mathbb{E}_D \lVert W_{2n} * \dots * W_1 x - y\rVert^2 + \lambda \sum_{i=1}^n \lVert W_{2i} \rVert_1$$<br />
<br />
where W are the weights and * is convolution. Here we have chosen the coefficient 0 for the L1 penalty associated with odd-numbered layers and the coefficient 1 for the L1 penalty associated with even-numbered layers. This loss is susceptible to trivial re-parameterizations: without affecting the least-squares loss, we can always reduce the LASSO loss by halving the weights of all even-numbered layers and doubling the weights of all odd-numbered layers.<br />
<br />
Furthermore, batch normalization (Ioffe, 2015) is incompatible with this method of weight regularization. Consider batch normalization at the <math>l</math>-th layer.<br />
<br />
<center><math>x^{l+1} = max\{\gamma \cdot BN_{\mu,\sigma,\epsilon}(W^l * x^l) + \beta, 0\}</math></center><br />
<br />
Due to the batch normalization, any uniform scaling of <math>W^l</math> which would change <math>l_1</math> and <math>l_2</math> norms, but has no have no effect on <math>x^{l+1}</math>. Thus, when trying to minimize weight norms of multiple layers, it is unclear how to properly choose penalties for each layer. Therefore, penalizing the norm of a filter in a deep convolutional network is hard to justify from a theoretical perspective.<br />
<br />
<br />
Thus, although not providing a complete theoretical guarantee on loss, Ye et al. (2018) develop a pruning technique that claims to be more justified than norm-based pruning is.<br />
<br />
== Method ==<br />
<br />
At a high level, Ye et al. (2018) propose that, instead of discovering sparsity via penalizing the per-filter or per-channel norm, penalize the batch normalization scale parameters ''gamma'' instead. The reasoning is that by having fewer parameters to constrain and working with normalized values, sparsity is easier to enforce, monitor, and learn. Having sparse batch normalization terms has the effect of pruning '''entire''' channels: if ''gamma'' is zero, then the output at that layer becomes constant (the bias term), and thus the preceding channels can be pruned.<br />
<br />
=== Summary ===<br />
<br />
The basic algorithm can be summarized as follows:<br />
<br />
1. Penalize the L1-norm of the batch normalization scaling parameters in the loss<br />
<br />
2. Train until loss plateaus<br />
<br />
3. Remove channels that correspond to a downstream zero in batch normalization<br />
<br />
4. Fine-tune the pruned model using regular learning<br />
<br />
=== Details ===<br />
<br />
There still exist a few problems that this summary has not addressed so far. Sub-gradient descent is known to have inverse square root convergence rate on subdifferentials (Gordon et al., 2012), so the sparsity gradient descent update may be suboptimal. Furthermore, the sparse penalty needs to be normalized with respect to previous channel sizes, since the penalty should be roughly equally distributed across all convolution layers.<br />
<br />
==== Slow Convergence ====<br />
To address the issue of slow convergence, Ye et al. (2018) use an iterative shrinking-thresholding algorithm (ISTA) (Beck & Teboulle, 2009) to update the batch normalization scale parameter. The intuition for ISTA is that the structure of the optimization objective can be taken advantage of. Consider: $$L(x) = f(x) + g(x).$$<br />
<br />
Let ''f'' be the model loss and ''g'' be the non-differentiable penalty (LASSO). ISTA is able to use the structure of the loss and converge in O(1/n), instead of O(1/sqrt(n)) when using subgradient descent, which assumes no structure about the loss. Even though ISTA is used in convex settings, Ye et. al (2018) argue that it still performs better than gradient descent.<br />
<br />
==== Penalty Normalization ====<br />
<br />
In the paper, Ye et al. (2018) normalize the per-layer sparse penalty with respect to the global input size, the current layer kernel areas, the previous layer kernel areas, and the local input feature map area.<br />
<br />
[[File:Screenshot_from_2018-02-28_17-06-41.png]] (Ye et al., 2018)<br />
<br />
To control the global penalty, a hyperparamter ''rho'' is multiplied with all the per-layer ''lambda'' in the final loss.<br />
<br />
=== Steps ===<br />
<br />
The final algorithm can be summarized as follows:<br />
<br />
1. Compute the per-layer normalized sparse penalty constant <math>\lambda</math><br />
<br />
2. Compute the global LASSO loss with global scaling constant <math>\rho</math><br />
<br />
3. Until convergence, train scaling parameters using ISTA and non-scaling parameters using regular gradient descent.<br />
<br />
4. Remove channels that correspond to a downstream zero in batch normalization<br />
<br />
5. Fine-tune the pruned model using regular learning<br />
<br />
== Results ==<br />
<br />
The authors show state-of-the-art performance, compared with other channel-pruning approaches. It is important to note that it would be unfair to compare against general pruning approaches; channel pruning specifically removes channels without introducing '''intra-kernel sparsity''', whereas other pruning approaches introduce irregular kernel sparsity and hence computational inefficiencies.<br />
<br />
=== CIFAR-10 Experiment ===<br />
<br />
[[File:Screenshot_from_2018-02-28_17-24-25.png]]<br />
<br />
For the convNet, reducing the number of parameters in the base model increased the accuracy in model A. This suggests that the base model is over-parameterized. Otherwise, there would be a trade-off of accuracy and model efficiency.<br />
<br />
=== ILSVRC2012 Experiment ===<br />
<br />
The authors note that while ResNet-101 takes hundreds of epochs to train, pruning only takes 5-10, with fine-tuning adding another 2, giving an empirical example how long pruning might take in practice.<br />
<br />
[[File:Screenshot_from_2018-02-28_17-24-36.png]]<br />
<br />
=== Image Foreground-Background Segmentation Experiment ===<br />
<br />
The authors note that it is common practice to take a network with pre-trained on a large task and fine-tune it to apply it to a different, smaller task. One might expect there might be some extra channels that while useful for the large task, can be omitted for the simpler task. This experiment replicated that use-case by taking a NN originally trained on multiple datasets and applying the proposed pruning method. The authors note that the pruned network actually improves over the original network in all but the most challenging test dataset, which is in line with the initial expectation. The results are shown in table below<br />
<br />
[[File:paper8_Segmentation.png|700px]]<br />
<br />
== Conclusion ==<br />
<br />
Pruning large neural architectures to fit on low-power devices is an important task. For a real quantitative measure of efficiency, it would be interesting to conduct actual power measurements on the pruned models versus baselines; reduction in FLOPs doesn't necessarily correspond with vastly reduced power since memory accesses dominate energy consumption (Han et al., 2015). However, the reduction in the number of FLOPs and parameters is encouraging, so moderate power savings should be expected.<br />
<br />
It would also be interesting to combine multiple approaches, or "throw the whole kitchen sink" at this task. Han et al. (2015) sparked much recent interest by successfully combining weight pruning, quantization, and Huffman coding without loss in accuracy. However, their approach introduced irregular sparsity in the convolutional layers, so a direct comparison cannot be made.<br />
<br />
In conclusion, this novel, theoretically-motivated interpretation of channel pruning was successfully applied to several important tasks.<br />
<br />
== References ==<br />
<br />
* Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2012). Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems (pp. 1097-1105).<br />
* He, K., Zhang, X., Ren, S., & Sun, J. (2016). Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 770-778).<br />
* Cheng, Y., Wang, D., Zhou, P., & Zhang, T. (2017). A Survey of Model Compression and Acceleration for Deep Neural Networks. arXiv preprint arXiv:1710.09282.<br />
* Ye, J., Lu, X., Lin, Z., & Wang, J. Z. (2018). Rethinking the Smaller-Norm-Less-Informative Assumption in Channel Pruning of Convolution Layers. arXiv preprint arXiv:1802.00124.<br />
* Li, H., Kadav, A., Durdanovic, I., Samet, H., & Graf, H. P. (2016). Pruning filters for efficient convnets. arXiv preprint arXiv:1608.08710.<br />
* Molchanov, P., Tyree, S., Karras, T., Aila, T., & Kautz, J. (2016). Pruning convolutional neural networks for resource efficient inference.<br />
* Ioffe, S., & Szegedy, C. (2015, June). Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International conference on machine learning (pp. 448-456).<br />
* Gordon, G., & Tibshirani, R. (2012). Subgradient method. https://www.cs.cmu.edu/~ggordon/10725-F12/slides/06-sg-method.pdf<br />
* Beck, A., & Teboulle, M. (2009). A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM journal on imaging sciences, 2(1), 183-202.<br />
* Han, S., Mao, H., & Dally, W. J. (2015). Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=36028stat946w18/AmbientGAN: Generative Models from Lossy Measurements2018-04-03T01:43:26Z<p>J3ku: Clarify Convolve + Noise experiment</p>
<hr />
<div>= Introduction =<br />
Generative Adversarial Networks operate by simulating complex distributions but training them requires access to large amounts of high quality data. Often times we only have access to noisy or partial observations, which will, from here on, be referred to as measurements of the true data. If we know the measurement function and would like to train a generative model for the true data, there are several ways to continue which have varying degrees of success. We will use noisy MNIST data as an illustrative example, and show the results of 1. ignoring the problem, 2. trying to recover the lost information, and 3. using AmbientGAN as a way to recover the true data distribution. Suppose we only see MNIST data that has been run through a Gaussian kernel (blurred) with some noise from a <math>N(0, 0.5^2)</math> distribution added to each pixel:<br />
<br />
<gallery mode="packed"><br />
File:mnist.png| True Data (Unobserved)<br />
File:mnistmeasured.png| Measured Data (Observed)<br />
</gallery><br />
<br />
<br />
=== Ignore the problem ===<br />
[[File:GANignore.png|500px]] [[File:mnistignore.png|300px]]<br />
<br />
Train a generative model directly on the measured data. This will obviously be unable to generate the true distribution before measurement has occurred. <br />
<br />
<br />
=== Try to recover the information lost ===<br />
[[File:GANrecovery.png|420px]] [[File:mnistrecover.png|300px]]<br />
<br />
Works better than ignoring the problem but depends on how easily the measurement function can be inverted.<br />
<br />
=== AmbientGAN ===<br />
[[File:GANambient.png|500px]] [[File:mnistambient.png|300px]]<br />
<br />
Ashish Bora, Eric Price and Alexandros G. Dimakis propose AmbientGAN as a way to recover the true underlying distribution from measurements of the true data. AmbientGAN works by training a generator which attempts to have the measurements of the output it generates fool the discriminator. The discriminator must distinguish between real and generated measurements. This paper is published in ICLR 2018.<br />
<br />
= Related Work = <br />
Currently there exist two distinct approaches for constructing neural network based generative models; they are autoregressive [4,5] and adversarial [6] based methods. The adversarial model has shown to be very successful in modeling complex data distributions such as images, 3D models, state action distributions and many more. This paper is related to the work in [7] where the authors create 3D object shapes from a dataset of 2D projections. This paper states that the work in [7] is a special case of the AmbientGAN framework where the measurement process creates 2D projections using weighted sums of voxel occupancies.<br />
<br />
= Datasets and Model Architectures=<br />
We used three datasets for our experiments: MNIST, CelebA and CIFAR-10 datasets We briefly describe the generative models used for the experiments. For the MNIST dataset, we use two GAN models. The first model is a conditional DCGAN, while the second model is an unconditional Wasserstein GAN with gradient penalty (WGANGP). For the CelebA dataset, we use an unconditional DCGAN. For the CIFAR-10 dataset, we use an Auxiliary Classifier Wasserstein GAN with gradient penalty (ACWGANGP). For measurements with 2D outputs, i.e. Block-Pixels, Block-Patch, Keep-Patch, Extract-Patch, and Convolve+Noise, we use the same discriminator architectures as in the original work. For 1D projections, i.e. Pad-Rotate-Project, Pad-Rotate-Project-θ, we use fully connected discriminators. The architecture of the fully connected discriminator used for the MNIST dataset was 25-25-1 and for the CelebA dataset was 100-100-1.<br />
<br />
= Model =<br />
For the following variables superscript <math>r</math> represents the true distributions while superscript <math>g</math> represents the generated distributions. Let <math>x</math>, represent the underlying space and <math>y</math> for the measurement.<br />
<br />
Thus, <math>p_x^r</math> is the real underlying distribution over <math>\mathbb{R}^n</math> that we are interested in. However if we assume that our (known) measurement functions, <math>f_\theta: \mathbb{R}^n \to \mathbb{R}^m</math> are parameterized by <math>\Theta \sim p_\theta</math>, we can then observe <math>Y = f_\theta(x) \sim p_y^r</math> where <math>p_y^r</math> is a distribution over the measurements <math>y</math>.<br />
<br />
Mirroring the standard GAN setup we let <math>Z \in \mathbb{R}^k, Z \sim p_z</math> and <math>\Theta \sim p_\theta</math> be random variables coming from a distribution that is easy to sample. <br />
<br />
If we have a generator <math>G: \mathbb{R}^k \to \mathbb{R}^n</math> then we can generate <math>X^g = G(Z)</math> which has distribution <math>p_x^g</math> a measurement <math>Y^g = f_\Theta(G(Z))</math> which has distribution <math>p_y^g</math>. <br />
<br />
Unfortunately we do not observe any <math>X^g \sim p_x</math> so we can use the discriminator directly on <math>G(Z)</math> to train the generator. Instead we will use the discriminator to distinguish between the <math>Y^g -<br />
f_\Theta(G(Z))</math> and <math>Y^r</math>. That is we train the discriminator, <math>D: \mathbb{R}^m \to \mathbb{R}</math> to detect if a measurement came from <math>p_y^r</math> or <math>p_y^g</math>.<br />
<br />
AmbientGAN has the objective function:<br />
<br />
\begin{align}<br />
\min_G \max_D \mathbb{E}_{Y^r \sim p_y^r}[q(D(Y^r))] + \mathbb{E}_{Z \sim p_z, \Theta \sim p_\theta}[q(1 - D(f_\Theta(G(Z))))]<br />
\end{align}<br />
<br />
where <math>q(.)</math> is the quality function; for the standard GAN <math>q(x) = log(x)</math> and for Wasserstein GAN <math>q(x) = x</math>.<br />
<br />
As a technical limitation we require <math>f_\theta</math> to be differentiable with the respect each input for all values of <math>\theta</math>.<br />
<br />
With this set up we sample <math>Z \sim p_z</math>, <math>\Theta \sim p_\theta</math>, and <math>Y^r \sim U\{y_1, \cdots, y_s\}</math> each iteration and use them to compute the stochastic gradients of the objective function. We alternate between updating <math>G</math> and updating <math>D</math>. <br />
<br />
= Empirical Results =<br />
<br />
The paper continues to present results of AmbientGAN under various measurement functions when compared to baseline models. We have already seen one example in the introduction: a comparison of AmbientGAN in the Convolve + Noise Measurement case compared to the ignore-baseline, and the unmeasure-baseline. <br />
<br />
=== Convolve + Noise ===<br />
Additional results with the convolve + noise case with the celebA dataset. The AmbientGAN is compared to the baseline results with Wiener deconvolution. It is clear that AmbientGAN has superior performance in this case. The measurement is created using a Gaussian kernel and IID Gaussian noise, with <math>f_{\Theta}(x) = k*x + \Theta</math>, where <math>*</math> is the convolution operation, <math>k</math> is the convolution kernel, and <math>\Theta \sim p_{\theta}</math> is the noise distribution.<br />
<br />
[[File:paper7_fig3.png]]<br />
<br />
Images undergone convolve + noise transformations (left). Results with Wiener deconvolution (middle). Results with AmbientGAN (right).<br />
<br />
=== Block-Pixels ===<br />
With the block-pixels measurement function each pixel is independently set to 0 with probability <math>p</math>.<br />
<br />
[[File:block-pixels.png]]<br />
<br />
Measurements from the celebA dataset with <math>p=0.95</math> (left). Images generated from GAN trained on unmeasured (via blurring) data (middle). Results generated from AmbientGAN (right).<br />
<br />
=== Block-Patch ===<br />
<br />
[[File:block-patch.png]]<br />
<br />
A random 14x14 patch is set to zero (left). Unmeasured using-navier-stoke inpainting (middle). AmbientGAN (right). <br />
<br />
=== Pad-Rotate-Project-<math>\theta</math> ===<br />
<br />
[[File:pad-rotate-project-theta.png]]<br />
<br />
Results generated by AmbientGAN where the measurement function 0 pads the images, rotates it by <math>\theta</math>, and projects it on to the x axis. For each measurement the value of <math>\theta</math> is known. <br />
<br />
The generated images only have the basic features of a face and is referred to as a failure case in the paper. However the measurement function performs relatively well given how lossy the measurement function is. <br />
<br />
=== Explanation of Inception Score ===<br />
To evaluate GAN performance, the authors make use of the inception score, a metric introduced by Salimans et al.(2016). To evaluate the inception score on a datapoint, a pre-trained inception classification model (Szegedy et al. 2016) is applied to that datapoint, and the KL divergence between its label distribution conditional on the datapoint and its marginal label distribution is computed. This KL divergence is the inception score. The idea is that meaningful images should be recognized by the inception model as belonging to some class, and so the conditional distribution should have low entropy, while the model should produce a variety of images, so the marginal should have high entropy. Thus an effective GAN should have a high inception score.<br />
<br />
=== MNIST Inception ===<br />
<br />
[[File:MNIST-inception.png]]<br />
<br />
AmbientGAN was compared with baselines through training several models with different probability <math>p</math> of blocking pixels. The plot on the left shows that the inception scores change as the block probability <math>p</math> changes. All four models are similar when no pixels are blocked <math>(p=0)</math>. By the increase of the blocking probability, AmbientGAN models present a relatively stable performance and perform better than the baseline models. Therefore, AmbientGAN is more robust than all other baseline models.<br />
<br />
The plot on the right reveals the changes in inception scores while the standard deviation of the additive Gaussian noise increased. Baselines perform better when the noise is small. By the increase of the variance, AmbientGAN models present a much better performance compare to the baseline models. Further AmbientGAN retains high inception scores as measurements become more and more lossy.<br />
<br />
For 1D projection, Pad-Rotate-Project model achieved an inception score of 4.18. Pad-Rotate-Project-θ model achieved an inception score of 8.12, which is close to the score of vanilla GAN 8.99.<br />
<br />
=== CIFAR-10 Inception ===<br />
<br />
[[File:CIFAR-inception.png]]<br />
<br />
AmbientGAN is faster to train and more robust even on more complex distributions such as CIFAR-10. Similar trends were observed on the CIFAR-10 data, and AmbientGAN maintains relatively stable inception score as the block probability was increased.<br />
<br />
=== Robustness To Measurement Model ===<br />
<br />
In order to empirically gauge robustness to measurement modelling error, the authors used the block-pixels measurement model: the image dataset was computed with <math> p^* = 0.5 </math>, and several versions of the model were trained, each using different values of blocking probability <math> p </math>. The inception scores were calculated and plotted as a function of <math> p </math>. This is shown on the left below:<br />
<br />
[[File:robustnessambientgan.png | 800px]]<br />
<br />
The authors observe that the inception score peaks when the model uses the correct probability, but decreases smoothly as the probability moves away, demonstrating some robustness.<br />
<br />
= Theoretical Results =<br />
<br />
The theoretical results in the paper prove the true underlying distribution of <math>p_x^r</math> can be recovered when we have data that comes from the Gaussian-Projection measurement, Fourier transform measurement and the block-pixels measurement. The do this by showing the distribution of the measurements <math>p_y^r</math> corresponds to a unique distribution <math>p_x^r</math>. Thus even when the measurement itself is non-invertible the effect of the measurement on the distribution <math>p_x^r</math> is invertible. Lemma 5.1 ensures this is sufficient to provide the AmbientGAN training process with a consistency guarantee. For full proofs of the results please see appendix A. <br />
<br />
=== Lemma 5.1 === <br />
Let <math>p_x^r</math> be the true data distribution, and <math>p_\theta</math> be the distributions over the parameters of the measurement function. Let <math>p_y^r</math> be the induced measurement distribution. <br />
<br />
Assume for <math>p_\theta</math> there is a unique probability distribution <math>p_x^r</math> that induces <math>p_y^r</math>. <br />
<br />
Then for the standard GAN model if the discriminator <math>D</math> is optimal such that <math>D(\cdot) = \frac{p_y^r(\cdot)}{p_y^r(\cdot) + p_y^g(\cdot)}</math>, then a generator <math>G</math> is optimal if and only if <math>p_x^g = p_x^r</math>. <br />
<br />
=== Theorems 5.2===<br />
For the Gussian-Projection measurement model, there is a unique underlying distribution <math>p_x^{r} </math> that can induce the observed measurement distribution <math>p_y^{r} </math>.<br />
<br />
=== Theorems 5.3===<br />
Let <math> \mathcal{F} (\cdot) </math> denote the Fourier transform and let <math>supp (\cdot) </math> be the support of a function. Consider the Convolve+Noise measurement model with the convolution kernel <math> k </math>and additive noise distribution <math>p_\theta </math>. If <math> supp( \mathcal{F} (k))^{c}=\phi </math> and <math> supp( \mathcal{F} (p_\theta))^{c}=\phi </math>, then there is a unique distribution <math>p_x^{r} </math> that can induce the measurement distribution <math>p_y^{r} </math>.<br />
<br />
=== Theorems 5.4===<br />
Assume that each image pixel takes values in a finite set P. Thus <math>x \in P^n \subset \mathbb{R}^{n} </math>. Assume <math>0 \in P </math>, and consider the Block-Pixels measurement model with <math>p </math> being the probability of blocking a pixel. If <math>p <1</math>, then there is a unique distribution <math>p_x^{r} </math> that can induce the measurement distribution <math>p_y^{r} </math>. Further, for any <math> \epsilon > 0, \delta \in (0, 1] </math>, given a dataset of<br />
\begin{equation}<br />
s=\Omega \left( \frac{|P|^{2n}}{(1-p)^{2n} \epsilon^{2}} log \left( \frac{|P|^{n}}{\delta} \right) \right)<br />
\end{equation}<br />
IID measurement samples from pry , if the discriminator D is optimal, then with probability <math> \geq 1 - \delta </math> over the dataset, any optimal generator G must satisfy <math> d_{TV} \left( p^g_x , p^r_x \right) \leq \epsilon </math>, where <math> d_{TV} \left( \cdot, \cdot \right) </math> is the total variation distance.<br />
<br />
= Conclusion =<br />
Generative models are powerful tools, but constructing a generative model requires a large, high quality dataset of the distribution of interest. The authors show how to relax this requirement, by learning a distribution from a dataset that only contains incomplete, noisy measurements of the distribution. This allows for the construction of new generative models of distributions for which no high quality dataset exists.<br />
<br />
= Future Research =<br />
<br />
One critical weakness of AmbientGAN is the assumption that the measurement model is known. It would be nice to be able to train an AmbientGAN model when we have an unknown measurement model but also a small sample of unmeasured data.<br />
<br />
A related piece of work is [https://arxiv.org/abs/1802.01284 here]. In particular, Algorithm 2 in the paper excluding the discriminator is similar to AmbientGAN.<br />
<br />
= References =<br />
# https://openreview.net/forum?id=Hy7fDog0b<br />
# Salimans, Tim, et al. "Improved techniques for training gans." Advances in Neural Information Processing Systems. 2016.<br />
# Szegedy, Christian, et al. "Rethinking the inception architecture for computer vision." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2016.<br />
# Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv:1312.6114, 2013.<br />
# Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016a.<br />
# Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural infor- mation processing systems, pp. 2672–2680, 2014.<br />
# Matheus Gadelha, Subhransu Maji, and Rui Wang. 3d shape induction from 2d views of multiple objects. arXiv preprint arXiv:1612.05872, 2016.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=One-Shot_Imitation_Learning&diff=36022One-Shot Imitation Learning2018-04-03T01:27:57Z<p>J3ku: Grammar and spelling edits</p>
<hr />
<div>= Introduction =<br />
Robotic systems can be used for many applications, but to truly be useful for complex applications, they need to overcome 2 challenges: having the intent of the task at hand communicated to them, and being able to perform the manipulations necessary to complete this task. It is preferable to use demonstration to teach the robotic systems rather than natural language, as natural language may often fail to convey the details and intricacies required for the task. However, current work on learning from demonstrations is only successful with large amounts of feature engineering or a large number of demonstrations. The proposed model aims to achieve 'one-shot' imitation learning, ie. learning to complete a new task from just a single demonstration of it without any other supervision. As input, the proposed model takes the observation of the current instance of a task, and a demonstration of successfully solving a different instance of the same task. Strong generalization was achieved by using a soft attention mechanism on both the sequence of actions and states that the demonstration consists of, as well as on the vector of element locations within the environment. The success of this proposed model at completing a series of block stacking tasks can be viewed at http://bit.ly/nips2017-oneshot.<br />
<br />
= Related Work =<br />
While one-shot imitation learning is a novel combination of ideas, each of the components has previously been studied.<br />
* Imitation Learning: <br />
** Behavioural learning uses supervised learning to map from observations to actions (e.g. [https://papers.nips.cc/paper/95-alvinn-an-autonomous-land-vehicle-in-a-neural-network.pdf (Pomerleau 1988)], [https://arxiv.org/pdf/1011.0686.pdf (Ross et. al 2011)])<br />
** Inverse reinforcement learning estimates a reward function that considers demonstrations as optimal behavior (e.g. [http://ai.stanford.edu/~ang/papers/icml00-irl.pdf (Ng et. al 2000)])<br />
* One-Shot Learning:<br />
** Typically a form of meta-learning<br />
** Previously used for variety of tasks but all domain-specific<br />
** [https://arxiv.org/abs/1703.03400 (Finn et al. 2017)] proposed a generic solution but excluded imitation learning<br />
* Reinforcement Learning:<br />
** Demonstrated to work on variety of tasks and environments, in particular on games and robotic control<br />
** Requires large amount of trials and a user-specified reward function<br />
* Multi-task/Transfer Learning:<br />
** Shown to be particularly effective at computer vision tasks<br />
** Not meant for one-shot learning<br />
* Attention Modelling:<br />
** The proposed model makes use of the attention model from [https://arxiv.org/abs/1409.0473 (Bahdanau et al. 2016)]<br />
** The attention modelling over demonstration is similar in nature to the seq2seq models from the well known [https://papers.nips.cc/paper/5346-sequence-to-sequence-learning-with-neural-networks.pdf (Sutskever et al. 2014)]<br />
<br />
= One-Shot Imitation Learning =<br />
<br />
[[File:oneshot1.jpg|1000px]]<br />
<br />
The figure above shows the differences between traditional and one-shot imitation learning. In a), the traditional method may require training different policies for performing similar tasks that are similar in nature. For example, stacking blocks to a height of 2 and to a height of 3. In b), the one-shot imitation learning allows the same policy to be used for these tasks given a single demonstration, achieving good performance without any additional system interactions. In c), the policy is trained by using a set of different training tasks, with enough examples so that the learned results can be generalized to other similar tasks. Each task has a set of successful demonstrations. Each iteration of training uses two demonstrations from a task, one is used as the input passing into the algorithm and the other is used at the output, the results from the two are then conditioned to produce the correct action.<br />
<br />
== Problem Formalization ==<br />
The problem is briefly formalized with the authors describing a distribution of tasks, an individual task, a distribution of demonstrations for this task, and a single demonstration respectively as \[T, t\sim T, D(t), d\sim D(t)\]<br />
In addition, an action, an observation, parameters, and a policy are respectively defined as \[a, o, \theta, \pi_\theta(a|o,d)\]<br />
In particular, a demonstration is a sequence of observation and action pairs \[d = [(o_1, a_1),(o_2, a_2), . . . ,(o_T , a_T )]\]<br />
Assuming that $$T$$ and some evaluation function $$R_t(d): R^T \rightarrow R$$ are given, and that succesful demonstrations are available for each task, then the objective is to maximize expectation of the policy performance over \[t\sim T, d\sim D(t)\].<br />
<br />
== Block Stacking Tasks ==<br />
The tasks that the authors focus on is block stacking. A user specifies in what final configuration cubic blocks should be stacked, and the goal is to use a 7-DOF Fetch robotic arm to arrange the blocks in this configuration. The number of blocks, and their desired configuration (ie. number of towers, the height of each tower, and order of blocks within each tower) can be varied and encoded as a string. For example, 'abc def' would signify 2 towers of height 3, with block A on block B on block C in one tower, and block D on block E on block F in a second tower. To add complexity, the initial configuration of the blocks can vary and is encoded as a set of 3-dimensional vectors describing the position of each block relative to the robotic arm.<br />
<br />
== Algorithm ==<br />
To avoid needing to specify a reward function, the authors use behavioral cloning and DAGGER, 2 imitation learning methods that require only demonstrations, for training. In each training step, a list of tasks is sampled, and for each, a demonstration with injected noise along with some observation-action pairs are sampled. Given the current observation and demonstration as input, the policy is trained against the sampled actions by minimizing L2 norm for continuous actions, and cross-entropy for discrete ones. Adamax is used as the optimizer with a learning rate of 0.001.<br />
<br />
= Architecture =<br />
The authors propose a novel architecture for imitation learning, consisting of 3 networks.<br />
<br />
[[File:oneshot2.jpg|1000px]]<br />
<br />
== Demonstration Network ==<br />
This network takes a demonstration as input and produces an embedding with size linearly proportional to the number of blocks and the size of the demonstration.<br />
=== Temporal Dropout ===<br />
Since a demonstration for block stacking can be very long, the authors randomly discard 95% of the time steps, a process they call 'temporal dropout'. The reduced size of the demonstrations allows multiple trajectories to be explored during testing to calculate an ensemble estimate. Dilated temporal convolutions and neighborhood attention are then repeatedly applied to the downsampled demonstrations.<br />
<br />
=== Neighborhood Attention ===<br />
Since demonstration sizes can vary, a mechanism is needed that is not restricted to fixed-length inputs. While soft attention is one such mechanism, the problem with it is that there may be increasingly large amounts of information lost if soft attention is used to map longer demonstrations to the same fixed length as shorter demonstrations. As a solution, the authors propose having the same number of outputs as inputs, but with attention performed on other inputs relative to the current input.<br />
<br />
A query <math>q</math>, a list of context vectors <math>\{c_j\}</math>, and a list of memory vectors <math>\{m_j\}</math> are given as input to soft attention. Each attention weight is given by the product of a learned weight vector and a nonlinearity applied to the sum of the query and corresponding context vector. Softmaxed weights applied to the corresponding memory vector form the output of the soft attention.<br />
<br />
\[Inputs: q, \{c_j\}, \{m_j\}\]<br />
\[Weights: w_i \leftarrow v^Ttanh(q+c_i)\]<br />
\[Output: \sum_i{m_i\frac{\exp(w_i)}{\sum_j{\exp(w_j)}}}\]<br />
<br />
A list of same-length embeddings, coming from a previous neighbourhood attention layer or a projection from the list of block coordinates, is given as input to neighborhood attention. For each block, two separate linear layers produce a query vector and a context vector, while a memory vector is a list of tuples that describe the position of each block joined with the input embedding for that block. Soft attention is then performed on this query, context vector, and memory vector. The authors claim that the intuition behind this process is to allow each block to provide information about itself relative to the other blocks in the environment. Finally, for each block, a linear transformation is performed on the vector composed by concatenating the input embedding, the result of the soft attention for that block, and the robot's state.<br />
<br />
For an environment with B blocks:<br />
\[State: s\]<br />
\[Block_i: b_i \leftarrow (x_i, y_i, z_i)\]<br />
\[Embeddings: h_1^{in}, ..., h_B^{in}\] <br />
\[Query_i: q_i \leftarrow Linear(h_i^{in})\]<br />
\[Context_i: c_i \leftarrow Linear(h_i^{in})\]<br />
\[Memory_i: m_i \leftarrow (b_i, h_i^{in}) \]<br />
\[Result_i: result_i \leftarrow SoftAttn(q_i, \{c_j\}_{j=1}^B, \{m_k\}_{k=1}^B)\]<br />
\[Output_i: output_i \leftarrow Linear(concat(h_i^{in}, result_i, b_i, s))\]<br />
<br />
== Context network ==<br />
This network takes the current state and the embedding produced by the demonstration network as inputs and outputs a fixed-length "context embedding" which captures only the information relevant for the manipulation network at this particular step.<br />
=== Attention over demonstration ===<br />
The current state is used to compute a query vector which is then used for attending over all the steps of the embedding. Since at each time step there are multiple blocks, the weights for each are summed together to produce a scalar for each time step. Neighbourhood attention is then applied several times, using an LSTM with untied weights, since the information at each time steps needs to be propagated to each block's embedding. <br />
<br />
Performing attention over the demonstration yields a vector whose size is independent of the demonstration size; however, it is still dependent on the number of blocks in the environment, so it is natural to now attend over the state in order to get a fixed-length vector.<br />
=== Attention over current state ===<br />
The authors propose that in general, within each subtask, only a limited number of blocks are relevant for performing the subtask. If the subtask is to stack A on B, then intuitively, one would suppose that only block A and B are relevant, and perhaps any blocks that may be blocking access to either A or B. This is not enforced during training, but once soft attention is applied to the current state to produce a fixed-length context embedding, the authors believe that the model does indeed learn in this way.<br />
<br />
== Manipulation network ==<br />
Given the context embedding as input, this simple feedforward network decides on the particular action needed, to complete the subtask of stacking one particular 'source' block on top of another 'target' block.<br />
<br />
= Experiments = <br />
The proposed model was tested on the block stacking tasks. the experiments were designed at answering the following questions:<br />
* How does training with behavioral cloning compare with DAGGER?<br />
* How does conditioning on the entire demonstration compare to conditioning on the final state?<br />
* How does conditioning on the entire demonstration compare to conditioning on a “snapshot” of the trajectory?<br />
* Can the authors' framework generalize to tasks that it has never seen during training?<br />
For the experiments, 140 training tasks and 43 testing tasks were collected, each with between 2 to 10 blocks and a different, desired final layout. Over 1000 demonstrations for each task were collected using a hard-coded policy rather than a human user. The authors compare 4 different architectures in these experiments:<br />
* Behavioural cloning used to train the proposed model<br />
* DAGGER used to train the proposed model<br />
* The proposed model, trained with DAGGER, but conditioned on the desired final state rather than an entire demonstration<br />
* The proposed model, trained with DAGGER, but conditioned on a 'snapshot' of the environment at the end of each subtask (ie. every time a block is stacked on another block)<br />
<br />
== Performance Evaluation ==<br />
[[File:oneshot3.jpg|1000px]]<br />
<br />
The most confident action at each timestep is chosen in 100 different task configurations, and results are averaged over tasks that had the same number of blocks. The results suggest that the performance of each of the architectures is comparable to that of the hard-coded policy which they aim to imitate. Performance degrades similarly across all architectures and the hard-coded policy as the number of blocks increases. On the harder tasks, conditioning on the entire demonstration led to better performance than conditioning on snapshots or on the final state. The authors believe that this may be due to the lack of information when conditioning only on the final state as well as due to regularization caused by temporal dropout which leads to data augmentation when conditioning on the full demonstration but is omitted when conditioning only on the snapshots or final state. Both DAGGER and behavioral cloning performed comparably well. As mentioned above, noise injection was used in training to improve performance; in practice, additional noise can still be injected but some may already come from other sources.<br />
<br />
== Visualization ==<br />
The authors visualize the attention mechanisms underlying the main policy architecture to have a better understanding about how it operates. There are two kinds of attention that the authors are mainly interested in, one where the policy attends to different time steps in the demonstration, and the other where the policy attends to different blocks in the current state. The figures below show some of the policy attention heatmaps over time.<br />
<br />
[[File:paper6_Visualization.png|800px]]<br />
<br />
= Conclusions =<br />
The proposed model successfully learns to complete new instances of a new task from just a single demonstration. The model was demonstrated to work on a series of block stacking tasks. The authors propose several extensions including enabling few-shot learning when one demonstration is insufficient, using image data as the demonstrations, and attempting many other tasks aside from block stacking.<br />
<br />
= Criticisms =<br />
While the paper shows an incredibly impressive result: the ability to learn a new task from just a single demonstration, there are a few points that need clearing up.<br />
Firstly, the authors use a hard-coded policy in their experiments rather than a human. It is clear that the performance of this policy begins to degrade quickly as the complexity of the task increases. It would be useful to know what this hard-coded policy actually was, and if the proposed model could still have comparable performance if a more successful demonstration, perhaps one by a human user, were performed. Give the current popularity of adversarial examples, it would also be interesting to see the performance when conditioned on an "adversarial" demonstration, that achieves the correct final state, but intentionally performs complex or obfuscated steps to get there.<br />
Second, it would be useful to see the model's performance on a more complex family of tasks than block stacking, since although each block stacking task is slightly different, the differences may turn out be insignificant compared to other tasks that this model should work on if it is to be a general imitation learning architecture. Regardless, this work is a big step forward for imitation learning, permitting a wider range of tasks for which there is little training data and no reward function available, to still be successfully solved.<br />
<br />
= Illustrative Example: Particle Reaching =<br />
<br />
[[File:f1.png]]<br />
<br />
Figure 1: [Left] Agent, [Middle] Orange square is target, [Right] Green triangle is target [2].<br />
<br />
Another simple yet insightful example of the One-Shot Imitation Learning is the particle reaching problem which provides a relatively simple suite of tasks from which the network needs to solve an arbitrary one. The problem is formulated such that for each task: there is an agent which can move based on a 2D force vector, and n landmarks at varying 2D locations (n varies from task to task) with the goal of moving the agent to the specific landmark reached in the demonstration. This is illustrated in Figure 1. <br />
<br />
[[File:f2.png|450px]]<br />
<br />
Figure 2: Experimental results [2].<br />
<br />
Some insight comes from the use of different network architectures to solve this problem. The three architectures to compare (described below) are plain LSTM, LSTM with attention, and final state with attention. The key insight is that the architectures go from generic to specific, with the best generalization performance achieved with the most specific architecture, final state with attention, as seen in Figure 2. It is important to note that this conclusion does not carry forward to more complicated tasks such as the block stacking task.<br />
*Plain LSTM: 512 hidden units, with the input being the demonstration trajectory (the position of the agent changes over time and approaches one of the targets). Output of the LSTM with the current state (from the task needed to be solved) is the input for a multi-layer perceptron (MLP) for finding the solution.<br />
*LSTM with attention: Output of LSTM is now a set of weights for the different targets during training. These weights and the test state are used in the test task. The, now, 2D output is the input for an MLP as before.<br />
*Final state with attention: Looks only at the final state of the demonstration since it can sufficiently provide the needed detail of which target to reach (trajectory is not required). Similar to previous architecture, produces weights used by MLP.<br />
<br />
= Source =<br />
# Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
# Duan, Yan, Marcin Andrychowicz, Bradly Stadie, OpenAI Jonathan Ho, Jonas Schneider, Ilya Sutskever, Pieter Abbeel, and Wojciech Zaremba. "One-shot imitation learning." In Advances in neural information processing systems, pp. 1087-1098. 2017.<br />
# Y. Duan, M. Andrychowicz, B. Stadie, J. Ho, J. Schneider, I. Sutskever, P. Abbeel, and W. Zaremba. One-shot imitation learning. arXiv preprint arXiv:1703.07326, 2017. (Newer revision)<br />
# Finn, Chelsea, Pieter Abbeel, and Sergey Levine. "Model-agnostic meta-learning for fast adaptation of deep networks." arXiv preprint arXiv:1703.03400 (2017).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=One-Shot_Imitation_Learning&diff=36021One-Shot Imitation Learning2018-04-03T01:14:31Z<p>J3ku: /* Related Work */ Add References</p>
<hr />
<div>= Introduction =<br />
Robotic systems can be used for many applications, but to truly be useful for complex applications, they need to overcome 2 challenges: having the intent of the task at hand communicated to them, and being able to perform the manipulations necessary to complete this task. It is preferable to use demonstration to teach the robotic systems rather than natural language, as natural language may often fail to convey the details and intricacies required for the task. However, current work on learning from demonstrations is only successful with large amounts of feature engineering or a large number of demonstrations. The proposed model aims to achieve 'one-shot' imitation learning, ie. learning to complete a new task from just a single demonstration of it without any other supervision. As input, the proposed model takes the observation of the current instance a task, and a demonstration of successfully solving a different instance of the same task. Strong generalization was achieved by using a soft attention mechanism on both the sequence of actions and states that the demonstration consists of, as well as on the vector of element locations within the environment. The success of this proposed model at completing a series of block stacking tasks can be viewed at http://bit.ly/nips2017-oneshot.<br />
<br />
= Related Work =<br />
While one-shot imitation learning is a novel combination of ideas, each of the components has previously been studied.<br />
* Imitation Learning: <br />
** Behavioural learning uses supervised learning to map from observations to actions (e.g. [https://papers.nips.cc/paper/95-alvinn-an-autonomous-land-vehicle-in-a-neural-network.pdf (Pomerleau 1988)], [https://arxiv.org/pdf/1011.0686.pdf (Ross et. al 2011)])<br />
** Inverse reinforcement learning estimates a reward function that considers demonstrations as optimal behavior (e.g. [http://ai.stanford.edu/~ang/papers/icml00-irl.pdf (Ng et. al 2000)])<br />
* One-Shot Learning:<br />
** Typically a form of meta-learning<br />
** Previously used for variety of tasks but all domain-specific<br />
** [https://arxiv.org/abs/1703.03400 (Finn et al. 2017)] proposed a generic solution but excluded imitation learning<br />
* Reinforcement Learning:<br />
** Demonstrated to work on variety of tasks and environments, in particular on games and robotic control<br />
** Requires large amount of trials and a user-specified reward function<br />
* Multi-task/Transfer Learning:<br />
** Shown to be particularly effective at computer vision tasks<br />
** Not meant for one-shot learning<br />
* Attention Modelling:<br />
** The proposed model makes use of the attention model from [https://arxiv.org/abs/1409.0473 (Bahdanau et al. 2016)]<br />
** The attention modelling over demonstration is similar in nature to the seq2seq models from the well known [https://papers.nips.cc/paper/5346-sequence-to-sequence-learning-with-neural-networks.pdf (Sutskever et al. 2014)]<br />
<br />
= One-Shot Imitation Learning =<br />
<br />
[[File:oneshot1.jpg|1000px]]<br />
<br />
The figure shoulds the differences between traditional and one-shot imitation learning. In a), the traditional method may require training different policies for performing similar tasks that are similar in nature. For example, stacking blocks in a height of 2 and in a height of 3. In b), the one-shot imitation learning allows the same policy to be used for these tasks given a single demonstration, achieving good performance without any additional system interactions. In c), the policy is trained by using a set of different training tasks, with enough examples so that the learned results can be generalized to other similar tasks. Each task has a set of successful demonstrations. Each iteration of training uses two demonstrations from a task, one is used as the input passing into the algorithm and the other is used at the output, the results from the two are then conditioned to produce the correct action.<br />
<br />
== Problem Formalization ==<br />
The problem is briefly formalized with the authors describing a distribution of tasks, an individual task, a distribution of demonstrations for this task, and a single demonstration respecitvely as \[T, t\sim T, D(t), d\sim D(t)\]<br />
In addition, an action, an observation, parameters, and a policy are respectively defined as \[a, o, \theta, \pi_\theta(a|o,d)\]<br />
In particular, a demonstration is a sequence of observation and action pairs \[d = [(o_1, a_1),(o_2, a_2), . . . ,(o_T , a_T )]\]<br />
Assuming that $$T$$ and some evaluation function $$R_t(d): R^T \rightarrow R$$ are given, and that succesful demonstrations are available for each task, then the objective is to maximize expectation of the policy performance over \[t\sim T, d\sim D(t)\].<br />
<br />
== Block Stacking Tasks ==<br />
The tasks that the authors focus on is block stacking. A user specifies in what final configuration cubic blocks should be stacked, and the goal is to use a 7-DOF Fetch robotic arm to arrange the blocks in this configuration. The number of blocks, and their desired configuration (ie. number of towers, the height of each tower, and order of blocks within each tower) can be varied and encoded as a string. For example, 'abc def' would signify 2 towers of height 3, with block A on block B on block C in one tower, and block D on block E on block F in a second tower. To add complexity, the initial configuration of the blocks can vary and is encoded as a set of 3-dimensional vectors describing the position of each block relative to the robotic arm.<br />
<br />
== Algorithm ==<br />
To avoid needing to specify a reward function, the authors use behavioral cloning and DAGGER, 2 imitation learning methods that require only demonstrations, for training. In each training step, a list of tasks is sampled, and for each, a demonstration with injected noise along with some observation-action pairs are sampled. Given the current observation and demonstration as input, the policy is trained against the sampled actions by minimizing L2 norm for continuous actions, and cross-entropy for discrete ones. Adamax is used as the optimizer with a learning rate of 0.001.<br />
<br />
= Architecture =<br />
The authors propose a novel architecture for imitation learning, consisting of 3 networks.<br />
<br />
[[File:oneshot2.jpg|1000px]]<br />
<br />
== Demonstration Network ==<br />
This network takes a demonstration as input and produces an embedding with size linearly proportional to the number of blocks and the size of the demonstration.<br />
=== Temporal Dropout: ===<br />
Since a demonstration for block stacking can be very long, the authors randomly discard 95% of the time steps, a process they call 'temporal dropout'. The reduced size of the demonstrations allows multiple trajectories to be explored during testing to calculate an ensemble estimate. Dilated temporal convolutions and neighborhood attention are then repeatedly applied to the downsampled demonstrations.<br />
<br />
=== Neighborhood Attention: ===<br />
Since demonstration sizes can vary, a mechanism is needed that is not restricted to fixed-length inputs. While soft attention is one such mechanism, the problem with it is that there may be increasingly large amounts of information lost if soft attention is used to map longer demonstrations to the same fixed length as shorter demonstrations. As a solution, the authors propose having the same number of outputs as inputs, but with attention performed on other inputs relative to the current input.<br />
<br />
A query <math>q</math>, a list of context vectors <math>\{c_j\}</math>, and a list of memory vectors <math>\{m_j\}</math> are given as input to soft attention. Each attention weight is given by the product of a learned weight vector and a nonlinearity applied to the sum of the query and corresponding context vector. Softmaxed weights applied to the corresponding memory vector form the output of the soft attention.<br />
<br />
\[Inputs: q, \{c_j\}, \{m_j\}\]<br />
\[Weights: w_i \leftarrow v^Ttanh(q+c_i)\]<br />
\[Output: \sum_i{m_i\frac{\exp(w_i)}{\sum_j{\exp(w_j)}}}\]<br />
<br />
A list of same-length embeddings, coming from a previous neighbourhood attention layer or a projection from the list of block coordinates, is given as input to neighborhood attention. For each block, two separate linear layers produce a query vector and a context vector, while a memory vector is a list of tuples that describe the position of each block joined with the input embedding for that block. Soft attention is then performed on this query, context vector, and memory vector. The authors claim that the intuition behind this process is to allow each block to provide information about itself relative to the other blocks in the environment. Finally, for each block, a linear transformation is performed on the vector composed by concatenating the input embedding, the result of the soft attention for that block, and the robot's state.<br />
<br />
For an environment with B blocks:<br />
\[State: s\]<br />
\[Block_i: b_i \leftarrow (x_i, y_i, z_i)\]<br />
\[Embeddings: h_1^{in}, ..., h_B^{in}\] <br />
\[Query_i: q_i \leftarrow Linear(h_i^{in})\]<br />
\[Context_i: c_i \leftarrow Linear(h_i^{in})\]<br />
\[Memory_i: m_i \leftarrow (b_i, h_i^{in}) \]<br />
\[Result_i: result_i \leftarrow SoftAttn(q_i, \{c_j\}_{j=1}^B, \{m_k\}_{k=1}^B)\]<br />
\[Output_i: output_i \leftarrow Linear(concat(h_i^{in}, result_i, b_i, s))\]<br />
<br />
== Context network ==<br />
This network takes the current state and the embedding produced by the demonstration network as inputs and outputs a fixed-length "context embedding" which captures only the information relevant for the manipulation network at this particular step.<br />
=== Attention over demonstration: ===<br />
The current state is used to compute a query vector which is then used for attending over all the steps of the embedding. Since at each time step there are multiple blocks, the weights for each are summed together to produce a scalar for each time step. Neighbourhood attention is then applied several times, using an LSTM with untied weights, since the information at each time steps needs to be propagated to each block's embedding. <br />
<br />
Performing attention over the demonstration yields a vector whose size is independent of the demonstration size; however, it is still dependent on the number of blocks in the environment, so it is natural to now attend over the state in order to get a fixed-length vector.<br />
=== Attention over current state: ===<br />
The authors propose that in general, within each subtask, only a limited number of blocks are relevant for performing the subtask. If the subtask is to stack A on B, then intuitively, one would suppose that only block A and B are relevant, and perhaps any blocks that may be blocking access to either A or B. This is not enforced during training, but once soft attention is applied to the current state to produce a fixed-length context embedding, the authors believe that the model does indeed learn in this way.<br />
<br />
== Manipulation network ==<br />
Given the context embedding as input, this simple feedforward network decides on the particular action needed, to complete the subtask of stacking one particular 'source' block on top of another 'target' block.<br />
<br />
= Experiments = <br />
The proposed model was tested on the block stacking tasks. the experiments were designed at answering the following questions:<br />
* How does training with behavioral cloning compare with DAGGER?<br />
* How does conditioning on the entire demonstration compare to conditioning on the final state?<br />
* How does conditioning on the entire demonstration compare to conditioning on a “snapshot” of the trajectory?<br />
* Can the authors' framework generalize to tasks that it has never seen during training?<br />
For the experiments, 140 training tasks and 43 testing tasks were collected, each with between 2 to 10 blocks and a different, desired final layout. Over 1000 demonstrations for each task were collected using a hard-coded policy rather than a human user. The authors compare 4 different architectures in these experiments:<br />
* Behavioural cloning used to train the proposed model<br />
* DAGGER used to train the proposed model<br />
* The proposed model, trained with DAGGER, but conditioned on the desired final state rather than an entire demonstration<br />
* The proposed model, trained with DAGGER, but conditioned on a 'snapshot' of the environment at the end of each subtask (ie. every time a block is stacked on another block)<br />
<br />
== Performance Evaluation ==<br />
[[File:oneshot3.jpg|1000px]]<br />
<br />
The most confident action at each timestep is chosen in 100 different task configurations, and results are averaged over tasks that had the same number of blocks. The results suggest that the performance of each of the architectures is comparable to that of the hard-coded policy which they aim to imitate. Performance degrades similarly across all architectures and the hard-coded policy as the number of blocks increases. On the harder tasks, conditioning on the entire demonstration led to better performance than conditioning on snapshots or on the final state. The authors believe that this may be due to the lack of information when conditioning only on the final state as well as due to regularization caused by temporal dropout which leads to data augmentation when conditioning on the full demonstration but is omitted when conditioning only on the snapshots or final state. Both DAGGER and behavioral cloning performed comparably well. As mentioned above, noise injection was used in training to improve performance; in practice, additional noise can still be injected but some may already come from other sources.<br />
<br />
== Visualization ==<br />
The authors visualize the attention mechanisms underlying the main policy architecture to have a better understanding about how it operates. There are two kinds of attention that the authors are mainly interested in, one where the policy attends to different time steps in the demonstration, and the other where the policy attends to different blocks in the current state. Figure below shows some of the attention heatmaps.<br />
<br />
[[File:paper6_Visualization.png|800px]]<br />
<br />
= Conclusions =<br />
The proposed model successfully learns to complete new instances of a new task from just a single demonstration. The model was demonstrated to work on a series of block stacking tasks. The authors propose several extensions including enabling few-shot learning when one demonstration is insufficient, using image data as the demonstrations, and attempting many other tasks aside from block stacking.<br />
<br />
= Criticisms =<br />
While the paper shows an incredibly impressive result: the ability to learn a new task from just a single demonstration, there are a few points that need clearing up.<br />
Firstly, the authors use a hard-coded policy in their experiments rather than a human. It is clear that the performance of this policy begins to degrade quickly as the complexity of the task increases. It would be useful to know what this hard-coded policy actually was, and if the proposed model could still have comparable performance if a more successful demonstration, perhaps one by a human user, were performed. Give the current popularity of adversarial examples, it would also be interesting to see the performance when conditioned on an "adversarial" demonstration, that achieves the correct final state, but intentionally performs complex or obfuscated steps to get there.<br />
Second, it would be useful to see the model's performance on a more complex family of tasks than block stacking, since although each block stacking task is slightly different, the differences may turn out be insignificant compared to other tasks that this model should work on if it is to be a general imitation learning architecture. Regardless, this work is a big step forward for imitation learning, permitting a wider range of tasks for which there is little training data and no reward function available, to still be successfully solved.<br />
<br />
= Illustrative Example: Particle Reaching =<br />
<br />
[[File:f1.png]]<br />
<br />
Figure 1: [Left] Agent, [Middle] Orange square is target, [Right] Green triangle is target [2].<br />
<br />
Another simple yet insightful example of the One-Shot Imitation Learning is the particle reaching problem which provides a relatively simple suite of tasks from which the network needs to solve an arbitrary one. The problem is formulated such that for each task: there is an agent which can move based on a 2D force vector, and n landmarks at varying 2D locations (n varies from task to task) with the goal of moving the agent to the specific landmark reached in the demonstration. This is illustrated in Figure 1. <br />
<br />
[[File:f2.png|450px]]<br />
<br />
Figure 2: Experimental results [2].<br />
<br />
Some insight comes from the use of different network architectures to solve this problem. The three architectures to compare (described below) are plain LSTM, LSTM with attention, and final state with attention. The key insight is that the architectures go from generic to specific, with the best generalization performance achieved with the most specific architecture, final state with attention, as seen in Figure 2. It is important to note that this conclusion does not carry forward to more complicated tasks such as the block stacking task.<br />
*Plain LSTM: 512 hidden units, with the input being the demonstration trajectory (the position of the agent changes over time and approaches one of the targets). Output of the LSTM with the current state (from the task needed to be solved) is the input for a multi-layer perceptron (MLP) for finding the solution.<br />
*LSTM with attention: Output of LSTM is now a set of weights for the different targets during training. These weights and the test state are used in the test task. The, now, 2D output is the input for an MLP as before.<br />
*Final state with attention: Looks only at the final state of the demonstration since it can sufficiently provide the needed detail of which target to reach (trajectory is not required). Similar to previous architecture, produces weights used by MLP.<br />
<br />
= Source =<br />
# Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
# Duan, Yan, Marcin Andrychowicz, Bradly Stadie, OpenAI Jonathan Ho, Jonas Schneider, Ilya Sutskever, Pieter Abbeel, and Wojciech Zaremba. "One-shot imitation learning." In Advances in neural information processing systems, pp. 1087-1098. 2017.<br />
# Y. Duan, M. Andrychowicz, B. Stadie, J. Ho, J. Schneider, I. Sutskever, P. Abbeel, and W. Zaremba. One-shot imitation learning. arXiv preprint arXiv:1703.07326, 2017. (Newer revision)<br />
# Finn, Chelsea, Pieter Abbeel, and Sergey Levine. "Model-agnostic meta-learning for fast adaptation of deep networks." arXiv preprint arXiv:1703.03400 (2017).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35269MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T19:42:30Z<p>J3ku: /* Approach */</p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background.<br />
<br />
[[File:marrnet_intro_image.png|700px|thumb|center|Objects in real images. The appearance of the same shaped object varies based on colour, texture, lighting, background, etc. However, the 2.5D sketches (e.g. depth or normal maps) of the object remain constant, and can be seen as an abstraction of the object which is used to reconstruct the 3D shape.]]<br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images and also enforce the re projection consistency between the 3D shape and the estimated sketch. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape. MarrNet uses an encoder-decoder structure for the sub-components of the framework. <br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
[[File:2-5d_example.PNG|700px|thumb|center|Results from the paper: Learning Non-Lambertian Object Intrinsics across ShapeNet Categories. The results show that neural networks can be trained to recover 2.5D information from an image. The top row predicts the albedo and the bottom row predicts the shading. It can be observed that the results are still blurry and the fine details are not fully recovered.]]<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in the figure below. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[[File:marrnet_model_components.png|700px|thumb|center|MarrNet architecture. 2.5D sketches of normals, depths, and silhouette are first estimated. The sketches are then used to estimate the 3D shape. Finally, re-projection consistency is used to ensure consistency between the sketch and 3D output.]]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
[[File:marrnet_reprojection_consistency.png|700px|thumb|center|Reprojection consistency for voxels. Left and middle: criteria for depth and silhouettes. Right: criterion for surface normals]]<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets.<br />
<br />
== ShapeNet ==<br />
Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br />
<br />
=== Method ===<br />
MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br />
<br />
=== Results ===<br />
The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. Quantitatively, the full model also achieves 0.57 IoU, higher than the direct prediction baseline.<br />
<br />
[[File:marrnet_shapenet_results.png|700px|thumb|center|ShapeNet results.]]<br />
<br />
== PASCAL 3D+ ==<br />
Rough 3D models are provided from real-life images.<br />
<br />
=== Method ===<br />
Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br />
<br />
=== Results ===<br />
The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br />
<br />
[[File:marrnet_pascal_3d_ablation.png|600px|thumb|center|Ablation studies using the PASCAL 3D+ dataset.]]<br />
<br />
Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. However, the authors claim that the IoU metric is sub-optimal for three reasons. First, there is no emphasis on details since the metric prefers models that predict mean shapes consistently. Second, all possible scales are searched during the IoU computation, making it less efficient. Third, PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, and computing IoU with these shapes is not very informative. Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth. This shows how MarrNet produces nice shapes and also highlights the fact that ground truth shapes are not very good.<br />
<br />
[[File:human_studies.png|400px|thumb|center|Human preferences on chairs in PASCAL 3D+ (Xiang et al. 2014). The numbers show the percentage of how often humans prefered the 3D shape from DRC (state-of-the-art), MarrNet, or GT.]]<br />
<br />
<br />
[[File:marrnet_pascal_3d_drc_comparison.png|600px|thumb|center|Comparison between DRC and MarrNet results.]]<br />
<br />
Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br />
<br />
[[File:marrnet_pascal_3d_failure_cases.png|500px|thumb|center|Failure cases on PASCAL 3D+. The algorithm cannot recover thin structures.]]<br />
<br />
===<br />
<br />
== IKEA ==<br />
This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br />
<br />
=== Results ===<br />
Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br />
<br />
[[File:marrnet_ikea_results.png|700px|thumb|center|Results on chairs in the IKEA dataset, and comparison with 3D-VAE-GAN.]]<br />
<br />
== Other Data ==<br />
MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br />
<br />
[[File:marrnet_airplanes_and_cars.png|700px|thumb|center|Results on airplanes and cars from the PASCAL 3D+ dataset, and comparison with DRC.]]<br />
<br />
MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below.<br />
<br />
[[File:marrnet_multiple_categories.png|700px|thumb|center|Results when trained jointly on all three object categories (cars, airplanes, and chairs).]]<br />
<br />
= Commentary =<br />
Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to distill the useful information for more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps also allows for easier training and domain adaptation from synthetic data.<br />
<br />
As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br />
<br />
As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder.<br />
<br />
= Conclusion =<br />
The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.<br />
<br />
= References =<br />
# David Marr. Vision: A computational investigation into the human representation and processing of visual information. W. H. Freeman and Company, 1982.<br />
# Shubham Tulsiani, Tinghui Zhou, Alexei A Efros, and Jitendra Malik. Multi-view supervision for single-view reconstruction via differentiable ray consistency. In CVPR, 2017.<br />
# JiajunWu, Chengkai Zhang, Tianfan Xue,William T Freeman, and Joshua B Tenenbaum. Learning a Proba- bilistic Latent Space of Object Shapes via 3D Generative-Adversarial Modeling. In NIPS, 2016b.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35237MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T18:11:30Z<p>J3ku: /* References */</p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background.<br />
<br />
[[File:marrnet_intro_image.png|700px|thumb|center|Objects in real images. The appearance of the same shaped object varies based on colour, texture, lighting, background, etc. However, the 2.5D sketches (e.g. depth or normal maps) of the object remain constant, and can be seen as an abstraction of the object which is used to reconstruct the 3D shape.]]<br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[[File:marrnet_model_components.png|700px|thumb|center|MarrNet architecture. 2.5D sketches of normals, depths, and silhouette are first estimated. The sketches are then used to estimate the 3D shape. Finally, re-projection consistency is used to ensure consistency between the sketch and 3D output.]]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
[[File:marrnet_reprojection_consistency.png|700px|thumb|center|Reprojection consistency for voxels. Left and middle: criteria for depth and silhouettes. Right: criterion for surface normals]]<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets.<br />
<br />
== ShapeNet ==<br />
Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br />
<br />
=== Method ===<br />
MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br />
<br />
=== Results ===<br />
The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. Quantitatively, the full model also achieves 0.57 IoU, higher than the direct prediction baseline.<br />
<br />
[[File:marrnet_shapenet_results.png|700px|thumb|center|ShapeNet results.]]<br />
<br />
== PASCAL 3D+ ==<br />
Rough 3D models are provided from real-life images.<br />
<br />
=== Method ===<br />
Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br />
<br />
=== Results ===<br />
The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br />
<br />
[[File:marrnet_pascal_3d_ablation.png|600px|thumb|center|Ablation studies using the PASCAL 3D+ dataset.]]<br />
<br />
Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. However, the authors claim that the IoU metric is sub-optimal for three reasons. First, there is no emphasis on details since the metric prefers models that predict mean shapes consistently. Second, all possible scales are searched during the IoU computation, making it less efficient. Third, PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, and computing IoU with these shapes is not very informative. Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth.<br />
<br />
[[File:marrnet_pascal_3d_drc_comparison.png|600px|thumb|center|Comparison between DRC and MarrNet results.]]<br />
<br />
Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br />
<br />
[[File:marrnet_pascal_3d_failure_cases.png|500px|thumb|center|Failure cases on PASCAL 3D+. The algorithm cannot recover thin structures.]]<br />
<br />
== IKEA ==<br />
This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br />
<br />
=== Results ===<br />
Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br />
<br />
[[File:marrnet_ikea_results.png|700px|thumb|center|Results on chairs in the IKEA dataset, and comparison with 3D-VAE-GAN.]]<br />
<br />
== Other Data ==<br />
MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br />
<br />
[[File:marrnet_airplanes_and_cars.png|700px|thumb|center|Results on airplanes and cars from the PASCAL 3D+ dataset, and comparison with DRC.]]<br />
<br />
MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below.<br />
<br />
[[File:marrnet_multiple_categories.png|700px|thumb|center|Results when trained jointly on all three object categories (cars, airplanes, and chairs).]]<br />
<br />
= Commentary =<br />
Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to distill the useful information for more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps also allows for easier training and domain adaptation from synthetic data.<br />
<br />
As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br />
<br />
As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder.<br />
<br />
= Conclusion =<br />
The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.<br />
<br />
= References =<br />
# David Marr. Vision: A computational investigation into the human representation and processing of visual information. W. H. Freeman and Company, 1982.<br />
# Shubham Tulsiani, Tinghui Zhou, Alexei A Efros, and Jitendra Malik. Multi-view supervision for single-view reconstruction via differentiable ray consistency. In CVPR, 2017.<br />
# JiajunWu, Chengkai Zhang, Tianfan Xue,William T Freeman, and Joshua B Tenenbaum. Learning a Proba- bilistic Latent Space of Object Shapes via 3D Generative-Adversarial Modeling. In NIPS, 2016b.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35235MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T18:09:54Z<p>J3ku: /* References */</p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background.<br />
<br />
[[File:marrnet_intro_image.png|700px|thumb|center|Objects in real images. The appearance of the same shaped object varies based on colour, texture, lighting, background, etc. However, the 2.5D sketches (e.g. depth or normal maps) of the object remain constant, and can be seen as an abstraction of the object which is used to reconstruct the 3D shape.]]<br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[[File:marrnet_model_components.png|700px|thumb|center|MarrNet architecture. 2.5D sketches of normals, depths, and silhouette are first estimated. The sketches are then used to estimate the 3D shape. Finally, re-projection consistency is used to ensure consistency between the sketch and 3D output.]]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
[[File:marrnet_reprojection_consistency.png|700px|thumb|center|Reprojection consistency for voxels. Left and middle: criteria for depth and silhouettes. Right: criterion for surface normals]]<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets.<br />
<br />
== ShapeNet ==<br />
Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br />
<br />
=== Method ===<br />
MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br />
<br />
=== Results ===<br />
The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. Quantitatively, the full model also achieves 0.57 IoU, higher than the direct prediction baseline.<br />
<br />
[[File:marrnet_shapenet_results.png|700px|thumb|center|ShapeNet results.]]<br />
<br />
== PASCAL 3D+ ==<br />
Rough 3D models are provided from real-life images.<br />
<br />
=== Method ===<br />
Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br />
<br />
=== Results ===<br />
The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br />
<br />
[[File:marrnet_pascal_3d_ablation.png|600px|thumb|center|Ablation studies using the PASCAL 3D+ dataset.]]<br />
<br />
Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. However, the authors claim that the IoU metric is sub-optimal for three reasons. First, there is no emphasis on details since the metric prefers models that predict mean shapes consistently. Second, all possible scales are searched during the IoU computation, making it less efficient. Third, PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, and computing IoU with these shapes is not very informative. Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth.<br />
<br />
[[File:marrnet_pascal_3d_drc_comparison.png|600px|thumb|center|Comparison between DRC and MarrNet results.]]<br />
<br />
Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br />
<br />
[[File:marrnet_pascal_3d_failure_cases.png|500px|thumb|center|Failure cases on PASCAL 3D+. The algorithm cannot recover thin structures.]]<br />
<br />
== IKEA ==<br />
This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br />
<br />
=== Results ===<br />
Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br />
<br />
[[File:marrnet_ikea_results.png|700px|thumb|center|Results on chairs in the IKEA dataset, and comparison with 3D-VAE-GAN.]]<br />
<br />
== Other Data ==<br />
MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br />
<br />
[[File:marrnet_airplanes_and_cars.png|700px|thumb|center|Results on airplanes and cars from the PASCAL 3D+ dataset, and comparison with DRC.]]<br />
<br />
MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below.<br />
<br />
[[File:marrnet_multiple_categories.png|700px|thumb|center|Results when trained jointly on all three object categories (cars, airplanes, and chairs).]]<br />
<br />
= Commentary =<br />
Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to distill the useful information for more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps also allows for easier training and domain adaptation from synthetic data.<br />
<br />
As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br />
<br />
As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder.<br />
<br />
= Conclusion =<br />
The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.<br />
<br />
= References =<br />
David Marr. Vision: A computational investigation into the human representation and processing of visual information. W. H. Freeman and Company, 1982.<br />
Shubham Tulsiani, Tinghui Zhou, Alexei A Efros, and Jitendra Malik. Multi-view supervision for single-view reconstruction via differentiable ray consistency. In CVPR, 2017.<br />
JiajunWu, Chengkai Zhang, Tianfan Xue,William T Freeman, and Joshua B Tenenbaum. Learning a Proba- bilistic Latent Space of Object Shapes via 3D Generative-Adversarial Modeling. In NIPS, 2016b.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35234MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T18:08:18Z<p>J3ku: </p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background.<br />
<br />
[[File:marrnet_intro_image.png|700px|thumb|center|Objects in real images. The appearance of the same shaped object varies based on colour, texture, lighting, background, etc. However, the 2.5D sketches (e.g. depth or normal maps) of the object remain constant, and can be seen as an abstraction of the object which is used to reconstruct the 3D shape.]]<br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[[File:marrnet_model_components.png|700px|thumb|center|MarrNet architecture. 2.5D sketches of normals, depths, and silhouette are first estimated. The sketches are then used to estimate the 3D shape. Finally, re-projection consistency is used to ensure consistency between the sketch and 3D output.]]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
[[File:marrnet_reprojection_consistency.png|700px|thumb|center|Reprojection consistency for voxels. Left and middle: criteria for depth and silhouettes. Right: criterion for surface normals]]<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets.<br />
<br />
== ShapeNet ==<br />
Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br />
<br />
=== Method ===<br />
MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br />
<br />
=== Results ===<br />
The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. Quantitatively, the full model also achieves 0.57 IoU, higher than the direct prediction baseline.<br />
<br />
[[File:marrnet_shapenet_results.png|700px|thumb|center|ShapeNet results.]]<br />
<br />
== PASCAL 3D+ ==<br />
Rough 3D models are provided from real-life images.<br />
<br />
=== Method ===<br />
Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br />
<br />
=== Results ===<br />
The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br />
<br />
[[File:marrnet_pascal_3d_ablation.png|600px|thumb|center|Ablation studies using the PASCAL 3D+ dataset.]]<br />
<br />
Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. However, the authors claim that the IoU metric is sub-optimal for three reasons. First, there is no emphasis on details since the metric prefers models that predict mean shapes consistently. Second, all possible scales are searched during the IoU computation, making it less efficient. Third, PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, and computing IoU with these shapes is not very informative. Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth.<br />
<br />
[[File:marrnet_pascal_3d_drc_comparison.png|600px|thumb|center|Comparison between DRC and MarrNet results.]]<br />
<br />
Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br />
<br />
[[File:marrnet_pascal_3d_failure_cases.png|500px|thumb|center|Failure cases on PASCAL 3D+. The algorithm cannot recover thin structures.]]<br />
<br />
== IKEA ==<br />
This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br />
<br />
=== Results ===<br />
Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br />
<br />
[[File:marrnet_ikea_results.png|700px|thumb|center|Results on chairs in the IKEA dataset, and comparison with 3D-VAE-GAN.]]<br />
<br />
== Other Data ==<br />
MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br />
<br />
[[File:marrnet_airplanes_and_cars.png|700px|thumb|center|Results on airplanes and cars from the PASCAL 3D+ dataset, and comparison with DRC.]]<br />
<br />
MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below.<br />
<br />
[[File:marrnet_multiple_categories.png|700px|thumb|center|Results when trained jointly on all three object categories (cars, airplanes, and chairs).]]<br />
<br />
= Commentary =<br />
Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to distill the useful information for more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps also allows for easier training and domain adaptation from synthetic data.<br />
<br />
As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br />
<br />
As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder.<br />
<br />
= Conclusion =<br />
The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.<br />
<br />
= References =</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18&diff=35233stat946w182018-03-22T18:07:36Z<p>J3ku: /* Paper presentation */</p>
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{| class="wikitable"<br />
<br />
{| border="1" cellpadding="3"<br />
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|width="60pt"|Date<br />
|width="100pt"|Name <br />
|width="30pt"|Paper number <br />
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|width="30pt"|Link to the paper<br />
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|Feb 15 (example)||Ri Wang || ||Sequence to sequence learning with neural networks.||[http://papers.nips.cc/paper/5346-sequence-to-sequence-learning-with-neural-networks.pdf Paper] || [http://wikicoursenote.com/wiki/Stat946f15/Sequence_to_sequence_learning_with_neural_networks#Long_Short-Term_Memory_Recurrent_Neural_Network Summary]<br />
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|Feb 27 || || 1|| || || <br />
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|Feb 27 || || 3|| || || <br />
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|Mar 1 || Peter Forsyth || 4|| Unsupervised Machine Translation Using Monolingual Corpora Only || [https://arxiv.org/pdf/1711.00043.pdf Paper] || [[https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only Summary]]<br />
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|Mar 1 || wenqing liu || 5|| Spectral Normalization for Generative Adversarial Networks || [https://openreview.net/pdf?id=B1QRgziT- Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Spectral_normalization_for_generative_adversial_network Summary]<br />
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|Mar 1 || Ilia Sucholutsky || 6|| One-Shot Imitation Learning || [https://papers.nips.cc/paper/6709-one-shot-imitation-learning.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=One-Shot_Imitation_Learning Summary]<br />
|-<br />
|Mar 6 || George (Shiyang) Wen || 7|| AmbientGAN: Generative models from lossy measurements || [https://openreview.net/pdf?id=Hy7fDog0b Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements Summary]<br />
|-<br />
|Mar 6 || Raphael Tang || 8|| Rethinking the Smaller-Norm-Less-Informative Assumption in Channel Pruning of Convolutional Layers || [https://arxiv.org/pdf/1802.00124.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Rethinking_the_Smaller-Norm-Less-Informative_Assumption_in_Channel_Pruning_of_Convolutional_Layers Summary]<br />
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|Mar 6 ||Fan Xia || 9|| Word translation without parallel data ||[https://arxiv.org/pdf/1710.04087.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data Summary]<br />
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|Mar 8 || Alex (Xian) Wang || 10 || Self-Normalizing Neural Networks || [http://papers.nips.cc/paper/6698-self-normalizing-neural-networks.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Self_Normalizing_Neural_Networks Summary] <br />
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|Mar 8 || Michael Broughton || 11|| Convergence of Adam and beyond || [https://openreview.net/pdf?id=ryQu7f-RZ Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=On_The_Convergence_Of_ADAM_And_Beyond Summary] <br />
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|Mar 8 || Wei Tao Chen || 12|| Predicting Floor-Level for 911 Calls with Neural Networks and Smartphone Sensor Data || [https://openreview.net/forum?id=ryBnUWb0b Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Predicting_Floor-Level_for_911_Calls_with_Neural_Networks_and_Smartphone_Sensor_Data Summary]<br />
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|Mar 13 || Chunshang Li || 13 || UNDERSTANDING IMAGE MOTION WITH GROUP REPRESENTATIONS || [https://openreview.net/pdf?id=SJLlmG-AZ Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Understanding_Image_Motion_with_Group_Representations Summary] <br />
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|Mar 13 || Saifuddin Hitawala || 14 || Robust Imitation of Diverse Behaviors || [https://papers.nips.cc/paper/7116-robust-imitation-of-diverse-behaviors.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Robust_Imitation_of_Diverse_Behaviors Summary] <br />
|-<br />
|Mar 13 || Taylor Denouden || 15|| A neural representation of sketch drawings || [https://arxiv.org/pdf/1704.03477.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=A_Neural_Representation_of_Sketch_Drawings Summary]<br />
|-<br />
|Mar 15 || Zehao Xu || 16|| Synthetic and natural noise both break neural machine translation || [https://openreview.net/pdf?id=BJ8vJebC- Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Synthetic_and_natural_noise_both_break_neural_machine_translation Summary]<br />
|-<br />
|Mar 15 || Prarthana Bhattacharyya || 17|| Wasserstein Auto-Encoders || [https://arxiv.org/pdf/1711.01558.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Wasserstein_Auto-Encoders Summary] <br />
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|Mar 15 || Changjian Li || 18|| Label-Free Supervision of Neural Networks with Physics and Domain Knowledge || [https://arxiv.org/pdf/1609.05566.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Label-Free_Supervision_of_Neural_Networks_with_Physics_and_Domain_Knowledge Summary]<br />
|-<br />
|Mar 20 || Travis Dunn || 19|| Continuous Adaptation via Meta-Learning in Nonstationary and Competitive Environments || [https://openreview.net/pdf?id=Sk2u1g-0- Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Continuous_Adaptation_via_Meta-Learning_in_Nonstationary_and_Competitive_Environments Summary]<br />
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|Mar 20 || Sushrut Bhalla || 20|| MaskRNN: Instance Level Video Object Segmentation || [https://papers.nips.cc/paper/6636-maskrnn-instance-level-video-object-segmentation.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/MaskRNN:_Instance_Level_Video_Object_Segmentation Summary]<br />
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|Mar 20 || Hamid Tahir || 21|| Wavelet Pooling for Convolution Neural Networks || [https://openreview.net/pdf?id=rkhlb8lCZ Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Wavelet_Pooling_CNN Summary]<br />
|-<br />
|Mar 22 || Dongyang Yang|| 22|| Implicit Causal Models for Genome-wide Association Studies || [https://openreview.net/pdf?id=SyELrEeAb Paper] ||[https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Implicit_Causal_Models_for_Genome-wide_Association_Studies Summary]<br />
|-<br />
|Mar 22 || Yao Li || 23||Improving GANs Using Optimal Transport || [https://openreview.net/pdf?id=rkQkBnJAb Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/IMPROVING_GANS_USING_OPTIMAL_TRANSPORT Summary]<br />
|-<br />
|Mar 22 || Sahil Pereira || 24||End-to-End Differentiable Adversarial Imitation Learning|| [http://proceedings.mlr.press/v70/baram17a/baram17a.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=End-to-End_Differentiable_Adversarial_Imitation_Learning Summary]<br />
|-<br />
|Mar 27 || Jaspreet Singh Sambee || 25|| Do Deep Neural Networks Suffer from Crowding? || [http://papers.nips.cc/paper/7146-do-deep-neural-networks-suffer-from-crowding.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Do_Deep_Neural_Networks_Suffer_from_Crowding Summary]<br />
|-<br />
|Mar 27 || Braden Hurl || 26|| Spherical CNNs || [https://openreview.net/pdf?id=Hkbd5xZRb Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=Spherical_CNNs Summary]<br />
|-<br />
|Mar 27 || Marko Ilievski || 27|| Neural Audio Synthesis of Musical Notes with WaveNet Autoencoders || [http://proceedings.mlr.press/v70/engel17a/engel17a.pdf Paper] || <br />
|-<br />
|Mar 29 || Alex Pon || 28||PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space || [https://arxiv.org/abs/1706.02413 Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=PointNet%2B%2B:_Deep_Hierarchical_Feature_Learning_on_Point_Sets_in_a_Metric_Space Summary]<br />
|-<br />
|Mar 29 || Sean Walsh || 29||Multi-scale Dense Networks for Resource Efficient Image Classification || [https://arxiv.org/pdf/1703.09844.pdf Paper] ||<br />
|-<br />
|Mar 29 || Jason Ku || 30||MarrNet: 3D Shape Reconstruction via 2.5D Sketches || [https://arxiv.org/pdf/1711.03129.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches Summary] <br />
|-<br />
|Apr 3 || Tong Yang || 31|| Dynamic Routing Between Capsules. || [http://papers.nips.cc/paper/6975-dynamic-routing-between-capsules.pdf Paper] || <br />
|-<br />
|Apr 3 || Benjamin Skikos || 32|| Training and Inference with Integers in Deep Neural Networks || [https://openreview.net/pdf?id=HJGXzmspb Paper] || <br />
|-<br />
|Apr 3 || Weishi Chen || 33|| Tensorized LSTMs for Sequence Learning || [https://arxiv.org/pdf/1711.01577.pdf Paper] || [https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Tensorized_LSTMs&action=edit&redlink=1 Summary] <br />
|-</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35176MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T09:04:42Z<p>J3ku: /* Commentary */</p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background.<br />
<br />
[[File:marrnet_intro_image.png|700px|thumb|center|Objects in real images. The appearance of the same shaped object varies based on colour, texture, lighting, background, etc. However, the 2.5D sketches (e.g. depth or normal maps) of the object remain constant, and can be seen as an abstraction of the object which is used to reconstruct the 3D shape.]]<br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[[File:marrnet_model_components.png|700px|thumb|center|MarrNet architecture. 2.5D sketches of normals, depths, and silhouette are first estimated. The sketches are then used to estimate the 3D shape. Finally, re-projection consistency is used to ensure consistency between the sketch and 3D output.]]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
[[File:marrnet_reprojection_consistency.png|700px|thumb|center|Reprojection consistency for voxels. Left and middle: criteria for depth and silhouettes. Right: criterion for surface normals]]<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets.<br />
<br />
== ShapeNet ==<br />
Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br />
<br />
=== Method ===<br />
MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br />
<br />
=== Results ===<br />
The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. Quantitatively, the full model also achieves 0.57 IoU, higher than the direct prediction baseline.<br />
<br />
[[File:marrnet_shapenet_results.png|700px|thumb|center|ShapeNet results.]]<br />
<br />
== PASCAL 3D+ ==<br />
Rough 3D models are provided from real-life images.<br />
<br />
=== Method ===<br />
Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br />
<br />
=== Results ===<br />
The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br />
<br />
[[File:marrnet_pascal_3d_ablation.png|600px|thumb|center|Ablation studies using the PASCAL 3D+ dataset.]]<br />
<br />
Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. However, the authors claim that the IoU metric is sub-optimal for three reasons. First, there is no emphasis on details since the metric prefers models that predict mean shapes consistently. Second, all possible scales are searched during the IoU computation, making it less efficient. Third, PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, and computing IoU with these shapes is not very informative. Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth.<br />
<br />
[[File:marrnet_pascal_3d_drc_comparison.png|600px|thumb|center|Comparison between DRC and MarrNet results.]]<br />
<br />
Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br />
<br />
[[File:marrnet_pascal_3d_failure_cases.png|500px|thumb|center|Failure cases on PASCAL 3D+. The algorithm cannot recover thin structures.]]<br />
<br />
== IKEA ==<br />
This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br />
<br />
=== Results ===<br />
Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br />
<br />
[[File:marrnet_ikea_results.png|700px|thumb|center|Results on chairs in the IKEA dataset, and comparison with 3D-VAE-GAN.]]<br />
<br />
== Other Data ==<br />
MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br />
<br />
[[File:marrnet_airplanes_and_cars.png|700px|thumb|center|Results on airplanes and cars from the PASCAL 3D+ dataset, and comparison with DRC.]]<br />
<br />
MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below.<br />
<br />
[[File:marrnet_multiple_categories.png|700px|thumb|center|Results when trained jointly on all three object categories (cars, airplanes, and chairs).]]<br />
<br />
= Commentary =<br />
Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to distill the useful information for more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps also allows for easier training and domain adaptation from synthetic data.<br />
<br />
As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br />
<br />
As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder.<br />
<br />
= Conclusion =<br />
The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35175MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T09:03:35Z<p>J3ku: /* Evaluation */</p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background.<br />
<br />
[[File:marrnet_intro_image.png|700px|thumb|center|Objects in real images. The appearance of the same shaped object varies based on colour, texture, lighting, background, etc. However, the 2.5D sketches (e.g. depth or normal maps) of the object remain constant, and can be seen as an abstraction of the object which is used to reconstruct the 3D shape.]]<br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[[File:marrnet_model_components.png|700px|thumb|center|MarrNet architecture. 2.5D sketches of normals, depths, and silhouette are first estimated. The sketches are then used to estimate the 3D shape. Finally, re-projection consistency is used to ensure consistency between the sketch and 3D output.]]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
[[File:marrnet_reprojection_consistency.png|700px|thumb|center|Reprojection consistency for voxels. Left and middle: criteria for depth and silhouettes. Right: criterion for surface normals]]<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets.<br />
<br />
== ShapeNet ==<br />
Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br />
<br />
=== Method ===<br />
MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br />
<br />
=== Results ===<br />
The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. Quantitatively, the full model also achieves 0.57 IoU, higher than the direct prediction baseline.<br />
<br />
[[File:marrnet_shapenet_results.png|700px|thumb|center|ShapeNet results.]]<br />
<br />
== PASCAL 3D+ ==<br />
Rough 3D models are provided from real-life images.<br />
<br />
=== Method ===<br />
Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br />
<br />
=== Results ===<br />
The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br />
<br />
[[File:marrnet_pascal_3d_ablation.png|600px|thumb|center|Ablation studies using the PASCAL 3D+ dataset.]]<br />
<br />
Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. However, the authors claim that the IoU metric is sub-optimal for three reasons. First, there is no emphasis on details since the metric prefers models that predict mean shapes consistently. Second, all possible scales are searched during the IoU computation, making it less efficient. Third, PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, and computing IoU with these shapes is not very informative. Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth.<br />
<br />
[[File:marrnet_pascal_3d_drc_comparison.png|600px|thumb|center|Comparison between DRC and MarrNet results.]]<br />
<br />
Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br />
<br />
[[File:marrnet_pascal_3d_failure_cases.png|500px|thumb|center|Failure cases on PASCAL 3D+. The algorithm cannot recover thin structures.]]<br />
<br />
== IKEA ==<br />
This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br />
<br />
=== Results ===<br />
Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br />
<br />
[[File:marrnet_ikea_results.png|700px|thumb|center|Results on chairs in the IKEA dataset, and comparison with 3D-VAE-GAN.]]<br />
<br />
== Other Data ==<br />
MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br />
<br />
[[File:marrnet_airplanes_and_cars.png|700px|thumb|center|Results on airplanes and cars from the PASCAL 3D+ dataset, and comparison with DRC.]]<br />
<br />
MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below.<br />
<br />
[[File:marrnet_multiple_categories.png|700px|thumb|center|Results when trained jointly on all three object categories (cars, airplanes, and chairs).]]<br />
<br />
= Commentary =<br />
Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to provide a strong shape prior, which results in more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps allows for easier training and domain adaptation from synthetic data.<br />
<br />
As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br />
<br />
As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder.<br />
<br />
= Conclusion =<br />
The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:marrnet_multiple_categories.png&diff=35174File:marrnet multiple categories.png2018-03-22T09:02:33Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:marrnet_airplanes_and_cars.png&diff=35173File:marrnet airplanes and cars.png2018-03-22T09:01:13Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:marrnet_ikea_results.png&diff=35172File:marrnet ikea results.png2018-03-22T08:59:47Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:marrnet_pascal_3d_failure_cases.png&diff=35171File:marrnet pascal 3d failure cases.png2018-03-22T08:57:47Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:marrnet_pascal_3d_drc_comparison.png&diff=35170File:marrnet pascal 3d drc comparison.png2018-03-22T08:56:24Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:marrnet_pascal_3d_ablation.png&diff=35169File:marrnet pascal 3d ablation.png2018-03-22T08:54:55Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:marrnet_shapenet_results.png&diff=35168File:marrnet shapenet results.png2018-03-22T08:53:18Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35167MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T08:52:22Z<p>J3ku: /* Approach */</p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background.<br />
<br />
[[File:marrnet_intro_image.png|700px|thumb|center|Objects in real images. The appearance of the same shaped object varies based on colour, texture, lighting, background, etc. However, the 2.5D sketches (e.g. depth or normal maps) of the object remain constant, and can be seen as an abstraction of the object which is used to reconstruct the 3D shape.]]<br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[[File:marrnet_model_components.png|700px|thumb|center|MarrNet architecture. 2.5D sketches of normals, depths, and silhouette are first estimated. The sketches are then used to estimate the 3D shape. Finally, re-projection consistency is used to ensure consistency between the sketch and 3D output.]]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
[[File:marrnet_reprojection_consistency.png|700px|thumb|center|Reprojection consistency for voxels. Left and middle: criteria for depth and silhouettes. Right: criterion for surface normals]]<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets.<br />
<br />
== ShapeNet ==<br />
Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br />
<br />
=== Method ===<br />
MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br />
<br />
=== Results ===<br />
The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. Quantitatively, the full model also achieves 0.57 IoU, higher than the direct prediction baseline.<br />
<br />
== PASCAL 3D+ ==<br />
Rough 3D models are provided from real-life images.<br />
<br />
=== Method ===<br />
Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br />
<br />
=== Results ===<br />
The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br />
<br />
[Figure]<br />
<br />
Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. However, the authors claim that the IoU metric is sub-optimal for three reasons. First, there is no emphasis on details since the metric prefers models that predict mean shapes consistently. Second, all possible scales are searched during the IoU computation, making it less efficient. Third, PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, and computing IoU with these shapes is not very informative.<br />
<br />
Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth.<br />
<br />
[Table]<br />
<br />
Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br />
<br />
[Figure]<br />
<br />
== IKEA ==<br />
This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br />
<br />
=== Results ===<br />
Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br />
<br />
== Other Data ==<br />
MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br />
<br />
[Figure]<br />
<br />
MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below. <br />
<br />
= Commentary =<br />
Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to provide a strong shape prior, which results in more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps allows for easier training and domain adaptation from synthetic data.<br />
<br />
As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br />
<br />
As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder.<br />
<br />
= Conclusion =<br />
The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:marrnet_reprojection_consistency.png&diff=35166File:marrnet reprojection consistency.png2018-03-22T08:51:06Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:marrnet_model_components.png&diff=35165File:marrnet model components.png2018-03-22T08:48:01Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35164MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T08:46:31Z<p>J3ku: /* Introduction */</p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background.<br />
<br />
[[File:marrnet_intro_image.png|700px|thumb|center|Objects in real images. The appearance of the same shaped object varies based on colour, texture, lighting, background, etc. However, the 2.5D sketches (e.g. depth or normal maps) of the object remain constant, and can be seen as an abstraction of the object which is used to reconstruct the 3D shape.]]<br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[Figure]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets.<br />
<br />
== ShapeNet ==<br />
Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br />
<br />
=== Method ===<br />
MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br />
<br />
=== Results ===<br />
The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. Quantitatively, the full model also achieves 0.57 IoU, higher than the direct prediction baseline.<br />
<br />
== PASCAL 3D+ ==<br />
Rough 3D models are provided from real-life images.<br />
<br />
=== Method ===<br />
Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br />
<br />
=== Results ===<br />
The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br />
<br />
[Figure]<br />
<br />
Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. However, the authors claim that the IoU metric is sub-optimal for three reasons. First, there is no emphasis on details since the metric prefers models that predict mean shapes consistently. Second, all possible scales are searched during the IoU computation, making it less efficient. Third, PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, and computing IoU with these shapes is not very informative.<br />
<br />
Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth.<br />
<br />
[Table]<br />
<br />
Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br />
<br />
[Figure]<br />
<br />
== IKEA ==<br />
This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br />
<br />
=== Results ===<br />
Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br />
<br />
== Other Data ==<br />
MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br />
<br />
[Figure]<br />
<br />
MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below. <br />
<br />
= Commentary =<br />
Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to provide a strong shape prior, which results in more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps allows for easier training and domain adaptation from synthetic data.<br />
<br />
As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br />
<br />
As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder.<br />
<br />
= Conclusion =<br />
The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=File:marrnet_intro_image.png&diff=35163File:marrnet intro image.png2018-03-22T08:42:55Z<p>J3ku: </p>
<hr />
<div></div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35162MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T08:39:55Z<p>J3ku: </p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background. <br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[Figure]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets.<br />
<br />
== ShapeNet ==<br />
Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br />
<br />
=== Method ===<br />
MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br />
<br />
=== Results ===<br />
The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. Quantitatively, the full model also achieves 0.57 IoU, higher than the direct prediction baseline.<br />
<br />
== PASCAL 3D+ ==<br />
Rough 3D models are provided from real-life images.<br />
<br />
=== Method ===<br />
Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br />
<br />
=== Results ===<br />
The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br />
<br />
[Figure]<br />
<br />
Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. However, the authors claim that the IoU metric is sub-optimal for three reasons. First, there is no emphasis on details since the metric prefers models that predict mean shapes consistently. Second, all possible scales are searched during the IoU computation, making it less efficient. Third, PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, and computing IoU with these shapes is not very informative.<br />
<br />
Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth.<br />
<br />
[Table]<br />
<br />
Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br />
<br />
[Figure]<br />
<br />
== IKEA ==<br />
This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br />
<br />
=== Results ===<br />
Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br />
<br />
== Other Data ==<br />
MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br />
<br />
[Figure]<br />
<br />
MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below. <br />
<br />
= Commentary =<br />
Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to provide a strong shape prior, which results in more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps allows for easier training and domain adaptation from synthetic data.<br />
<br />
As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br />
<br />
As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder.<br />
<br />
= Conclusion =<br />
The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35161MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T08:37:40Z<p>J3ku: /* Training */</p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background. <br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[Figure]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
<br />
== ShapeNet ==<br />
=== Data ===<br />
=== Method ===<br />
=== Results ===<br />
<br />
== PASCAL 3D+ ==<br />
=== Data ===<br />
=== Method ===<br />
=== Results ===<br />
<br />
== IKEA ==<br />
=== Data ===<br />
=== Results ===<br />
<br />
== Other Data ==<br />
<br />
= Commentary =<br />
<br />
= Conclusion =</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35160MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T08:37:15Z<p>J3ku: /* Surface Normals */</p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background. <br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[Figure]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
<br />
= Evaluation =<br />
<br />
== ShapeNet ==<br />
=== Data ===<br />
=== Method ===<br />
=== Results ===<br />
<br />
== PASCAL 3D+ ==<br />
=== Data ===<br />
=== Method ===<br />
=== Results ===<br />
<br />
== IKEA ==<br />
=== Data ===<br />
=== Results ===<br />
<br />
== Other Data ==<br />
<br />
= Commentary =<br />
<br />
= Conclusion =</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35159MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T08:29:42Z<p>J3ku: </p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background. <br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth and surface normal maps, the author’s design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape.<br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, with the use of depth and silhouettes, as well as some papers enforcing differentiable 2D-3D constraints for joint training of deep networks. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in Figure 1. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step, shown in Figure 2, estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[Figure]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information. A ResNet-18 encoder-decoder network is used, with the encoder taking a 256 x 256 RGB image, producing 8 x 8 x 512 feature maps. The decoder is four sets of 5 x 5 convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem. The network architecture is inspired by the TL network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. The projected depth loss is defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
<br />
= Training =<br />
<br />
= Evaluation =<br />
<br />
== ShapeNet ==<br />
=== Data ===<br />
=== Method ===<br />
=== Results ===<br />
<br />
== PASCAL 3D+ ==<br />
=== Data ===<br />
=== Method ===<br />
=== Results ===<br />
<br />
== IKEA ==<br />
=== Data ===<br />
=== Results ===<br />
<br />
== Other Data ==<br />
<br />
= Commentary =<br />
<br />
= Conclusion =</div>J3kuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=35092MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-03-22T02:55:12Z<p>J3ku: </p>
<hr />
<div>= Introduction =</div>J3ku