Noisy Matrix Completion under Sparse Factor Models

TL;DR

This paper develops theoretical error bounds and practical algorithms for noisy matrix completion under sparse factor models, applicable to various noise types and observation regimes, with experimental validation.

Contribution

It introduces estimation error bounds and an algorithm for noisy matrix completion with sparse factors, extending to diverse noise models and observation types.

Findings

Error bounds for sparsity-regularized maximum likelihood estimators.

Algorithm based on alternating direction method of multipliers.

Experimental results support theoretical error analyses.

Abstract

This paper examines a general class of noisy matrix completion tasks where the goal is to estimate a matrix from observations obtained at a subset of its entries, each of which is subject to random noise or corruption. Our specific focus is on settings where the matrix to be estimated is well-approximated by a product of two (a priori unknown) matrices, one of which is sparse. Such structural models - referred to here as "sparse factor models" - have been widely used, for example, in subspace clustering applications, as well as in contemporary sparse modeling and dictionary learning tasks. Our main theoretical contributions are estimation error bounds for sparsity-regularized maximum likelihood estimators for problems of this form, which are applicable to a number of different observation noise or corruption models. Several specific implications are examined, including scenarios where observations are corrupted by additive Gaussian noise or additive heavier-tailed (Laplace) noise, Poisson-distributed observations, and highly-quantized (e.g., one-bit) observations. We also propose a simple algorithmic approach based on the alternating direction method of multipliers for these tasks, and provide experimental evidence to support our error analyses.