Exponential Family Matrix Completion Under Structural Constraints

Abstract

We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low–rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin–tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data–types, such as skewed–continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low–rank, such as block–sparsity, or a superposition structure of low–rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of \textit{exponential} family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer \mathcal{R}(.). We propose a simple convex regularized M–estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets.