Error bounds for maximum likelihood matrix completion under sparse factor models

Abstract

This paper examines a general class of matrix completion tasks where entry wise observations of the matrix are subject to random noise or corruption. Our particular focus here is on settings where the matrix to be estimated follows a sparse factor model, in the sense that it may be expressed as the product of two matrices, one of which is sparse. We analyze the performance of a sparsity-penalized maximum likelihood approach to such problems to provide a general-purpose estimation result applicable to any of a number of noise/corruption models, and describe its implications in two stylized scenarios — one characterized by additive Gaussian noise, and the other by highly-quantized one-bit observations. We also provide some supporting empirical evidence to validate our theoretical claims in the Gaussian setting.