Provable Low Rank Plus Sparse Matrix Separation Via Nonconvex Regularizers
This addresses a fundamental matrix separation problem in machine learning and signal processing with theoretical guarantees, though it builds incrementally on existing nonconvex regularization approaches.
The paper tackles the problem of recovering low-rank and sparse components from measurements without the estimator bias of convex relaxations or requiring prior knowledge of rank/sparsity, achieving provable error bounds for the alternating proximal gradient descent algorithm applied to sparse optimization, matrix completion, and robust PCA.
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.