Low-rank solutions of linear matrix equations via procrustes flow

Abstract

In this paper we study the problem of recovering an low-rank positive semidefinite matrix from linear measurements. Our algorithm, which we call Procrustes Flow, starts from an ini-tial estimate obtained by a thresholding scheme followed by gradient descent on a non-convex objective. We show that as long as the measurements obey a standard restricted isometry property, our algorithm converges to the unknown matrix at a geometric rate. In the case of Gaussian measurements, such convergence occurs for a n×n matrix of rank r when the number of measurements exceeds a constant times nr. 1