Provable Low Rank Plus Sparse Matrix Separation Via Nonconvex Regularizers

TL;DR

This paper introduces a nonconvex regularization approach for recovering low rank and sparse matrices, providing theoretical error bounds and broad applicability to problems like matrix completion and robust PCA.

Contribution

It develops a novel analysis of an alternating proximal gradient descent algorithm for nonconvex regularizers, reducing estimator bias and handling unknown rank or sparsity.

Findings

Error bounds for the proposed algorithm

Applicability to matrix completion and robust PCA

Improved recovery performance over convex methods

Abstract

This paper considers a large class of problems where we seek to recover a low rank matrix and/or sparse vector from some set of measurements. While methods based on convex relaxations suffer from a (possibly large) estimator bias, and other nonconvex methods require the rank or sparsity to be known a priori, we use nonconvex regularizers to minimize the rank and $l_0$ norm without the estimator bias from the convex relaxation. We present a novel analysis of the alternating proximal gradient descent algorithm applied to such problems, and bound the error between the iterates and the ground truth sparse and low rank matrices. The algorithm and error bound can be applied to sparse optimization, matrix completion, and robust principal component analysis as special cases of our results.