MATRIX DATA DEEP DECODER -GEOMETRIC LEARN-ING FOR STRUCTURED DATA COMPLETION

Abstract

In this work we present a fully convolutional end to end method to reconstruct corrupted sparse matrices of Non-Euclidean data.The classic example for such matrices is recommender systems matrices where the rows/columns represent items/users and the entries are ratings. The method we present is inspired by the surprising and spectacular success of methods like" deep image prior" and "deep decoder" for corrupted image completion. In sharp contrast to previous Matrix Completion methods wherein the latent matrix or its factors directly serve as the optimization variable, in the method we present, the matrix is parametrized as the weights of a graph neural network acting on a random noisy input. Then we are tuning the network parameters to get a result as close as possible to the initial sparse matrix (using it's factors) getting that way state of the art matrix completion result. In addition to the conceptual simplicity of our method, which is just non-Euclidean generalization of deep image priors, it holds less parameters then previously presented methods which makes the parameters more trackable and the method more computationally efficient and more applicable for the real world tasks.The method also achieves state-of-the-art results for the matrix completion task on the classical benchmarks in the field. The method also surprisingly shows that untrained convolutional neural network can use a good prior not only for image completion but also for Matrix Completion when redefined for graphs.

1. INTRODUCTION

Matrix completion (MC) consists of estimating the missing entries of an n × m matrix X (usually, of very big dimensions) given its measurements M on a (usually, very sparse) support Ω. An example of such matrices are signals on graphs/manifolds which are Non-Euclidean domains. The classical example of such data are recommender (recommendation) systems, where the ratings are signals on (user item) couple. The most known Matrix Completion problem is the Netflix problem, where a 1M $ prize was offered for the algorithm that can best predict user ratings in a dataset that contained 480k movies × 18k users (8.5B entries), with 0.011% known entries (Bell et al., 2009) . Many works focused on solutions for the MC problem. In brief, one wishes to obtain the matrix X given matrix M as the specified input on the support Ω. Then, formally the completion task amounts to the minimization problem X = argmin X A Ω • (X -M ) 2 F where A Ω is the observation mask matrix (filled with 1 where data exists in the original problem), • is the Hadamard product and .

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F is the Frobenius norms (Rennie & Srebro, 2005) . Different approaches where presented in order to fill in matrix X. Those approached included imposing different regularization (priors) on the matrix and its factors. The most prominent approach consists of imposing a low rank (Candès & Recht, 2009; Recht, 2009) on the matrix. Then, priors based on collaborative filtering (users/items rating patterns), content based filtering (user/items profile) (Ghassemi et al., 2018; Jain & Dhillon, 2013; Xu et al., 2013; Si et al., 2016) and their combinations. Then Geometric Matrix Completion approaches appeared (Li & Yeung, 2009; Rao et al., 2015; Cai et al., 2011) and proposed describing rows/column graphs which represent similarity, then encoding the structural (geometric) information of those graphs via graph Laplacian regularization (Belkin & Partha, 2002; Belkin & Niyogi, 2003) and imposing smoothness of the data in those graphs

