probabilistic Matrix Factorization

UNDER CONSTRUCTION

Introduction and Motivation (Netflix Dataset)

The problem provides an approach to collaborative filtering that can scale linearly and, thus, handle large datasets. The methods are discussed in the context of the Netflix dataet.

This dataset is [math]\displaystyle{ N \times M }[/math] matrix where each row corresponds to a user, each column to a movie and the entries to a rating. Ratings take on values [math]\displaystyle{ 1,...,K }[/math]. The difficulty in this problem arises from the fact that each user has only rated a subset of movies. If there is some latent rank-[math]\displaystyle{ D }[/math] structure to these ratings (that is, underlying variables that determine how much a user enjoys a movie) then it would beneficial to factor the matrix.

Formally, given a preference matrix [math]\displaystyle{ R }[/math] which is [math]\displaystyle{ N \times M }[/math], we wish to find an [math]\displaystyle{ N \times D }[/math] matrix [math]\displaystyle{ U^T }[/math] and an [math]\displaystyle{ D \times M }[/math] matrix [math]\displaystyle{ V }[/math]. [math]\displaystyle{ V }[/math] is a factor matrix (i.e. for the [math]\displaystyle{ D }[/math] latent variables, it gives the value for each movie) and [math]\displaystyle{ U^T }[/math] gives coefficients of these factors for each user. The actual "Prize" dataset has [math]\displaystyle{ N=480,189, M = 17,770 }[/math] with over 100 million of these entries having values.

Another difficulty that arises with this dataset is that it is unbalanced. There is much variance in user activity; thus, some rows are much sparser than others.

There are several existing ways to solve this problem but they often require approximations that are slow and, at times, inaccurate. Another instinctive approach might be to use SVD. However, since there is a lot of missing data in this case, the optimization problem that needs to be solved is not convex. Thus, existing methods do not scale well and often have lower accuracy in cases where a row is sparse.

The proposed solution is to use a probabilistic model.

Probabilistic Matrix Factorization

Given the preference matrix described above with entries [math]\displaystyle{ R_{ij} }[/math]. The aim is to find a factorization that minimizes the root mean squared error (RMSE) on the test set, an initial attempt is to use linear model where we assume that there is Gaussian noise in the data (Figure 1). Define [math]\displaystyle{ I_{ij} }[/math] to be 1 if [math]\displaystyle{ R_{ij} }[/math] is known and 0 otherwise. Further, let [math]\displaystyle{ \,\mathcal N(x|\mu,\sigma^2) = f_X(x) }[/math] with [math]\displaystyle{ X \sim \mathcal N(\mu,\sigma^2) }[/math]. Then, we can define a conditional probability of the ratings with hyperparameter [math]\displaystyle{ \sigma^2 }[/math]

[math]\displaystyle{ p(R|U,V,\sigma^2) = \prod_{i=1}^N \prod_{j=1}^M \left[ \mathcal N(R_{ij}|U_i^TV_j,\sigma^2)\right]^{I_{ij}}\,\,\,\,\,\,\,\,(1) }[/math]

and priors on [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] with hyperparameters [math]\displaystyle{ \sigma^2_U, \sigma^2_V }[/math]

[math]\displaystyle{ p(U|\sigma^2_U) = \prod_{i=1}^N \mathcal N(U_i|0,\sigma^2_U\textbf{I})\,\,\,\,\,\,\,\,p(V|\sigma^2_V) = \prod_{i=1}^M \mathcal N(V_i|0,\sigma^2_V\textbf{I})\,\,\,\,\,\,\,\,(2) }[/math]

Then we aim to maximize the log posterior over [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] [math]\displaystyle{ \ln p(U,V,\sigma^2,\sigma^2_V,\sigma^2_U) = - \frac{1}{2\sigma^2}\sum_{i=1}^N\sum_{j=1}^M I_{ij}(R_{ij} - U_i^TV_j)^2 - \frac{1}{2\sigma^2_U} \sum_{i=1}^N U_i^TU_i - \frac{1}{2\sigma^2_V} \sum_{i=1}^N V_i^TV_i }[/math][math]\displaystyle{ - \frac{1}{2}\left(\kappa\ln\sigma^2 + ND\ln\sigma^2_U + MD\ln\sigma^2_V\right) + C\,\,\,\,\,\,\,\,(4) }[/math]

where [math]\displaystyle{ \kappa }[/math] is the number of entries that are known and [math]\displaystyle{ C }[/math] is a constant independent of the parameters. We can fix the three variance hyperparameters as constants which reduces the optimization to the first three terms. Then defining [math]\displaystyle{ \lambda_M = \sigma^2/\sigma^2_M }[/math] for [math]\displaystyle{ M=U,V }[/math] and multiplying by [math]\displaystyle{ -\sigma^2\lt 0 }[/math] results in a goal of minimizing

[math]\displaystyle{ E = \frac{1}{2}\left(\sum_{i=1}^N\sum_{j=1}^M I_{ij}(R_{ij} - U_i^TV_j)^2 + \lambda_U \sum_{i=1}^N \|U_i \|^2_F + \lambda_V \sum_{i=1}^M \| V_i \|^2_F \right)\,\,\,\,\,\,\,(5) }[/math]

where [math]\displaystyle{ \|A \|^2_F = \sum_{i=1}^m\sum_{j=1}^n |a_{ij}|^2 }[/math] is the Frobenius norm. Note that if all values are known (i.e. [math]\displaystyle{ I_{ij} = 1 \,\, \forall(i,j) }[/math] ) then as [math]\displaystyle{ \sigma^2_U, \sigma^2_V \rightarrow \infty }[/math], [math]\displaystyle{ E }[/math] reduces to the objective function of SVD.

It is also important to note that this model can result in values greater than [math]\displaystyle{ K }[/math] which is the range of the ratings. To remedy this, map the ratings to [math]\displaystyle{ [0,1] }[/math] with the function [math]\displaystyle{ t(x) = (x-1)/(K-1) }[/math]. Consequently, in the optimization problem (5), replace [math]\displaystyle{ (U_i^TV_j) }[/math] with [math]\displaystyle{ g((U_i^TV_j)) }[/math] where g is the logistic function, [math]\displaystyle{ g(x) = (1+\exp(-x))^{-1} }[/math].

Finally, (5) can be minimized using the method of steepest descent which is linear in the number of observations.

Constrained Probabilistic Matrix Factorization

In the above PMF approach, if a row is very sparse (that is, a user has rated very movies) then the estimated values will just approximately be the mean rating by the other users. Thus, we need a way remedy this. This is done by introducing a constraint and modifying PMF to created a constrained PMF algorithm.

First, we define an indicator matrix I which is 1 for entries that are known and 0 for entries that are not. Recall that D is the estimated number of latent factors. Let W be a [math]\displaystyle{ D \times M }[/math] that measures similarity between movies and the latent variables.

Then for a user i we can define [math]\displaystyle{ U_i = Y_i + \frac{\sum_{k=1}^M I_{ik}W_k} }[/math]

This means that users with similar sets of watched movies will have similar priors for [math]\displaystyle{ U_i }[/math]. Constrained PMF differs from PMF with the second term here so that [math]\displaystyle{ U_i \neq Y_i }[/math]. So, replacing [math]\displaystyle{ U_i }[/math] in the distribution above gives [math]\displaystyle{ p(R|Y,V,W,\sigma^2) = \prod_{i=1}^N \prod_{j=1}^M \left[ \mathcal N(R_{ij}|g\left(\left[Y_i + \frac{\sum_{k=1}^M I_{ik}W_k}\right]^TV_j\right),\sigma^2)\right]^{I_{ij}}\,\,\,\,\,\,\,\,(1) }[/math]

where, again, g is the logistic function.

Again, we can transform this maximization into a minimization like with (5) and use gradient descent (linear in number of observations). There is a significant improvement in performance.

Experimental Results

Consider a subset of the Netflix data set with 50,000 users and 1850 movies, with 1,082,982 known values. To show the improvement of constrained PMF, more than half of the rows are sparse (less than 10 entries).