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 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.

20 pages, 9 figures

References in corpus (1)

• Restricted strong convexity and weighted matrix completion: Optimal bounds with noise

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• Unified View of Matrix Completion under General Structural Constraints
• A Universal Variance Reduction-Based Catalyst for Nonconvex Low-Rank Matrix Recovery
• Structured Matrix Recovery via the Generalized Dantzig Selector
• A Geometric View on Constrained M-Estimators
• Consistent Collective Matrix Completion under Joint Low Rank Structure
• Low-rank matrix recovery with non-quadratic loss: projected gradient method and regularity projection oracle