Geometric Analysis of Noisy Low-Rank Matrix Recovery in the Exact Parametrized and the Overparametrized Regimes

Abstract

The matrix sensing problem is an important low-rank optimization problem that has found a wide range of applications, such as matrix completion, phase synchornization/retrieval, robust principal component analysis (PCA), and power system state estimation. In this work, we focus on the general matrix sensing problem with linear measurements that are corrupted by random noise. We investigate the scenario where the search rank r is equal to the true rank [Formula: see text] of the unknown ground truth (the exact parametrized case), as well as the scenario where r is greater than [Formula: see text] (the overparametrized case). We quantify the role of the restricted isometry property (RIP) in shaping the landscape of the nonconvex factorized formulation and assisting with the success of local search algorithms. First, we develop a global guarantee on the maximum distance between an arbitrary local minimizer of the nonconvex problem and the ground truth under the assumption that the RIP constant is smaller than [Formula: see text]. We then present a local guarantee for problems with an arbitrary RIP constant, which states that any local minimizer is either considerably close to the ground truth or far away from it. More importantly, we prove that this noisy, overparametrized problem exhibits the strict saddle property, which leads to the global convergence of perturbed gradient descent algorithm in polynomial time. The results of this work provide a comprehensive understanding of the geometric landscape of the matrix sensing problem in the noisy and overparametrized regime. Funding: This work was supported by grants from the National Science Foundation, Office of Naval Research, Air Force Office of Scientific Research, and Army Research Office.