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Presented by

Grace Tompkins, Tatiana Krikella, Swaleh Hussain

Introduction

One of the fundamental challenges in machine learning and data science is dealing with missing and incomplete data. This paper proposes theoretically justified methodology for using incomplete data in neural networks, eliminating the need for direct completion of the data by imputation or other commonly used methods in existing literature. The authors propose identifying missing data points with a parametric density and then training it together with the rest of the network's parameters. The neuron's response at the first hidden layer is generalized by taking its expected value to process this probabilistic representation. This process is essentially calculating the average activation of the neuron over imputations drawn from the missing data's density. The proposed approach is advantageous as it has the ability to train neural networks using incomplete observations from datasets, which are ubiquitous in practice. This approach also requires minimal adjustments and modifications to existing architectures. Theoretical results of this study show that this process does not lead to a loss of information, while experimental results showed the practical uses of this methodology on several different types of networks.

Related Work

Currently, dealing with incomplete inputs in machine learning requires filling absent attributes based on complete, observed data. Two commonly used methods are mean imputation and k-NN imputation. Other methods for dealing with missing data involve training separate neural networks, extreme learning machines, and [math]k[/math]-nearest neighbours. Probabilistic models of incomplete data can also be built depending on the mechanism missingness (i.e. whether the data is Missing At Random (MAR), Missing Completely At Random (MCAR), or Missing Not At Random (MNAR)), which can be fed into a particular learning model. Further, the decision function can also be trained using available/visible inputs alone. Previous work using neural networks for missing data includes a paper by Bengio and Gringras [1] where the authors used recurrent neural networks with feedback into the input units to fill absent attributes solely to minimize the learning criterion. Goodfellow et. al. [2] also used neural networks by introducing a multi-prediction deep Boltzmann machine which could perform classification on data with missingness in the inputs.

Layer for Processing Missing Data

In this approach, the adaptation of a given neural network to incomplete data relies on two steps: the estimation of the missing data and the generalization of the neuron's activation.

Let [math](x,J)[/math] represent a missing data point, where [math]x \in \mathbb{R}^D [/math], and [math]J \subset {1,...,D} [/math] is a set of attributes with missing data.

For each missing point [math](x,J)[/math], define an affine subspace consisting of all points which coincide with [math]x[/math] on known coordinates [math]J'=\{1,…,N\}/J[/math]:

[math]S=Aff[x,J]=span(e_J) [/math]

where [math]e_J=[e_j]_{j\in J}[/math] and [math]e_j[/math] is the [math] j^{th}[/math] canonical vector in [math]\mathbb{R}^D [/math].

Assume that the missing data points come from the D-dimensional probability distribution, [math]F[/math]. In their approach, the authors assume that the data points follow a mixture of Gaussians (GMM) with diagonal covariance matrices. By choosing diagonal covariance matrices, the number of model parameters is reduced. To model the missing points [math](x,J)[/math], the density [math]F[/math] is restricted to the affine subspace [math]S[/math]. Thus, possible values of [math](x,J)[/math] are modelled using the conditional density [math]F_S: S \to \mathbb{R} [/math],

[math]F_S(x) = \begin{cases} \frac{1}{\int_{S} F(s) \,ds}F(x) & \text{if $x \in S$,} \\ 0 & \text{otherwise.} \end{cases} [/math]

To process the missing data by a neural network, the authors propose that only the first hidden layer needs modification. Specifically, they generalize the activation functions of all the neurons in the first hidden layer of the network to process the probability density functions representing the missing data points. For the conditional density function [math]F_S[/math], the authors define the generalized activation of a neuron [math]n: \mathbb{R}^D \to \mathbb{R}[/math] on [math]F_S [/math] as:

[math]n(F_S)=E[n(x)|x \sim F_S]=\int n(x)F_S(x) \,dx[/math],

provided that the expectation exists.

The following two theorems describe how to apply the above generalizations to both the ReLU and the RBF neurons, respectively.

Theorem 3.1 Let [math]F = \sum_i{p_iN(m_i, \Sigma_i)}[/math] be the mixture of (possibly degenerate) Gaussians. Given weights [math]w=(w_1, ..., w_D) \in \mathbb{R}^D,[/math][math] b \in \mathbb{R} [/math], we have

[math]\text{ReLU}_{w,b}(F)=\sum_i{p_iNR\big(\frac{w^{\top}m_i+b}{\sqrt{w^{\top}\Sigma_iw}}}\big)[/math]

where [math]NR(x)=\text{ReLU}[N(x,1)][/math] and [math]\text{ReLU}_{w,b}(x)=\text{max}(w^{\top}+b, 0)[/math], [math]w \in \mathbb{R}^D [/math] and [math] b \in \mathbb{R}[/math] is the bias.

Theorem 3.2 Let [math]F = \sum_i{p_iN(m_i, \Sigma_i)}[/math] be the mixture of (possibly degenerate) Gaussians and let the RBF unit be parametrized by [math]N(c, \Gamma) [/math]. We have:

[math]\text{RBF}_{c, \Gamma}(F) = \sum_{i=1}^k{p_iN(m_i-c, \Gamma+\Sigma_i)}(0)[/math].

In the case where the data set contains no missing values, the generalized neurons reduce to classical ones, since the distribution [math]F[/math] is only used to estimate possible values at missing attributes. However, if one wishes to use an incomplete data set in the testing stage, then an incomplete data set must be used to train the model.

[math] [/math]

Theoretical Analysis

The main theoretical results, which are summarized below, show that using generalized neuron's activation at the first layer does not lead to the loss of information.

Let the generalized response of a neuron [math]n: \mathbb{R}^D \rightarrow \mathbb{R}[/math] evaluated on a probability measure [math]\mu[/math] which is given by

[math]n(\mu) := \int n(x)d\mu(x)[/math]
.

Theorem 4.1 shows that a neural network with generalized ReLU units is able to identify any two probability measures. The proof presented by the authors uses the Universal Approximation Property (UAP), and is summarized as follows.


Theorem 4.1. Let [math]\mu[/math], [math]v[/math] be probabilistic measures satisfying [math]\int ||x|| d \mu(x) \lt \infty[/math]. If

[math]ReLU_{w,b}(\mu) = ReLU_{w,b}(\nu) \text{ for } w \in \mathbb{R}^D, b \in \mathbb{R}[/math]
then [math]\nu = \mu.[/math] Sketch of Proof Let [math]w \in \mathbb{R}^D[/math] be fixed and define the set
[math]F_w = \{p: \mathbb{R} \rightarrow \mathbb{R}: \int p(w^Tx)d\mu(x) = \int p(w^Tx)d\nu(x)\}.[/math]
The first step of the proof involves showing that [math]F_w[/math] contains all continuous and bounded functions. The authors show this by showing that a piecewise continuous function that is affine linear on specific intervals, [math]Q[/math], is in the set [math]F_w[/math]. This involves re-writing [math]Q[/math] as a sum of tent-like piecewise linear functions, [math]T[/math] and showing that [math]T \in F_w[/math] (since it is sufficient to only show [math]T \in F_w[/math]).

Next, the authors show that an arbitrary bounded continuous function [math]G[/math] is in [math]F_w[/math] by the Lebesgue dominated convergence theorem.

Then, as [math]cos(\cdot), sin(\cdot) \in F_w[/math], the function
[math]exp(ir) = cos(r) + sin(r) \in F_w[/math]
and we have the equality
[math]\int exp(iw^Tx)d\mu(x) = \int exp(iw^Tx)d\nu(x).[/math]
Since [math]w[/math] was arbitrarily chosen, we can conclude that [math]\mu = \nu[/math]

as the characteristic functions of the two measures coincide.


More general results can be obtained making stronger assumptions on the probability measures, for example if a given family of neurons satisfies UAP, then their generalization can identify any probability measure with compact support.

Experimental Results

The model was applied to three types of algorithms: an Autoencoder (AE), a multilayer perceptron, and a radial basis function network.

Autoencoder

Corrupted images were restored as a part of this experiment. Grayscale handwritten digits were obtained from the MNIST database. A 13 by 13 (169 pixels) square was removed from each 28 by 28 (784 pixels) image. The location of the square was uniformly sampled for each image. The autoencoder used included 5 hidden layers. The first layer used ReLU activation functions while the subsequent layers utilized sigmoids. The loss function was computed using pixels from outside the mask.

Popular imputation techniques were compared against the conducted experiment:

k-nn: Replaced missing features with the mean of respective features calculated using K nearest training samples. Here, K=5.

mean: Replaced missing features with the mean of respective features calculated using all incomplete training samples.

dropout: Dropped input neutrons with missing values.

Moreover, a context encoder (CE) was trained by replacing missing features with their means. Unlike mean imputation, the complete data was used in the training phase. The method under study performed better than the imputation methods inside and outside the mask. Additionally, the method under study outperformed CE based on the whole area and area outside the mask.

Multilayer Perceptron

A multilayer perceptron with 3 ReLU hidden layers was applied to a multi-class classification problem on the Epileptic Seizure Recognition (ESR) data set taken from [3]. Each 178-dimensional vector (out of 11500 samples) is the EEG recording of a given person for 1 second, categorized into one of 5 classes. To generate missing attributes, 25%, 50%, 75%, and 90% of observations were randomly removed. The aforementioned imputation methods were used in addition to Multiple Imputation by Chained Equation (mice) and a mixture of Gaussians (gmm). The former utilizes the conditional distribution of data by Markov chain Monte Carlo techniques to draw imputations. The latter replaces missing features with values sampled from GMM estimated from incomplete data using the EM algorithm.

Double 5-fold cross-validation was used to report classification results. The model under study outperformed classical imputation methods, which give reasonable results only for a low number of missing values. The method under study performs nearly as well as CE, even though CE had access to complete training data.

Radial Basis Function Network

RBFN can be considered as a minimal architecture implementing our model, which contains only one hidden layer. A cross-entropy function was applied to a softmax in the output layer. Two-class data sets retrieved from the UCI repository [4] with internally missing attributes were used. Since the classification is binary, two additional SVM kernel models which work directly with incomplete data without performing any imputations were included in the experiment:

geom: The objective function is based on the geometric interpretation of the margin and aims to maximize the margin of each sample in its own subspace [5].

karma: This algorithm iteratively tunes kernel classifier under low-rank assumptions [6].

The above SVM methods were combined with RBF kernel function. The number of RBF units was selected in the inner cross-validation from the range {25, 50, 75, 100}. Initial centers of RBFNs were randomly selected from training data while variances were samples from N(0,1) distribution. For SVM methods, the margin parameter [math]C[/math] and kernel radius [math]\gamma[/math] were selected from [math]\{2^k :k=−5,−3,...,9\}[/math] for both parameters. For karma, additional parameter [math]\gamma_{karma}[/math] was selected from the set [math]\{1, 2\}[/math].

The model under study outperformed imputation techniques in almost all cases. It partially confirms that the use of raw incomplete data in neural networks is a usually better approach than filling missing attributes before the learning process. Moreover, it obtained more accurate results than modified kernel methods, which directly work on incomplete data. The performance of the model was once again comparable to, and in some cases better than CE, which had access to the complete data.

Conclusion

The results with these experiments along with the theoretical results conclude that this novel approach for dealing with missing data through a modification of a neural network is beneficial and outperforms many existing methods. This approach, which utilizes representing missing data with a probability density function, allows a neural network to determine a more generalized and accurate response of the neuron.

Critiques

- A simulation study where the mechanism of missingness is known will be interesting to examine. Doing this will allow us to see when the proposed method is better than existing methods, and under what conditions.

- This method of imputing incomplete data has many limitations: in most cases when we have a missing data point we are facing a relatively small amount of data that does not require training of a neural network. For a large dataset, missing records does not seem to be very crucial because obtaining data will be relatively easier, and using an empirical way of imputing data such as a majority vote will be sufficient.

- An interesting application of this problem is in NLP. In NLP, especially Question Answering, there is a problem where a query is given and an answer must be retrieved, but the knowledge base is incomplete. There is therefore a requirement for the model to be able to infer information from the existing knowledge base in order to answer the question. Although this problem is a little more contrived than the one mentioned here, it is nevertheless similar in nature because it requires the ability to probabilistically determine some value which can then be used as a response.

- The experiments in this paper evaluate this method against low amounts of missing data. It would be interesting to see the properties of this imputation when a majority of the data is missing, and see if this method can outperform dropout training in this setting (dropout is known to be surprisingly robust even at high drop levels).

- This problem can possibly be applied to face recognition where given a blurry image of a person's face, the neural network can make the image clearer such that the face of the person would be visible for humans to see and also possible for the software to identify who the person is.

References

[1] Yoshua Bengio and Francois Gingras. Recurrent neural networks for missing or asynchronous data. In Advances in neural information processing systems, pages 395–401, 1996.

[2] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep learning. MIT press, 2016.

[3] Ralph G Andrzejak, Klaus Lehnertz, Florian Mormann, Christoph Rieke, Peter David, and Christian E Elger. Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state. Physical Review E, 64(6):061907, 2001.

[4] Arthur Asuncion and David J. Newman. UCI Machine Learning Repository, 2007.

[5] Gal Chechik, Geremy Heitz, Gal Elidan, Pieter Abbeel, and Daphne Koller. Max-margin classification of data with absent features. Journal of Machine Learning Research, 9:1–21, 2008.

[6] Elad Hazan, Roi Livni, and Yishay Mansour. Classification with low rank and missing data. In Proceedings of The 32nd International Conference on Machine Learning, pages 257–266, 2015.