Low-rank Solutions of Linear Matrix Equations via Procrustes Flow

TL;DR

This paper introduces Procrustes Flow, an algorithm for recovering low-rank matrices from linear measurements, demonstrating geometric convergence under certain measurement conditions, particularly for Gaussian measurements.

Contribution

The paper proposes a novel non-convex optimization algorithm with proven convergence guarantees for low-rank matrix recovery from linear measurements.

Findings

Converges geometrically under restricted isometry property

Requires measurements proportional to matrix dimensions and rank

Effective for Gaussian measurement models

Abstract

In this paper we study the problem of recovering a low-rank matrix from linear measurements. Our algorithm, which we call Procrustes Flow, starts from an initial 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 $n_1 \times n_2$ matrix of rank $r$ when the number of measurements exceeds a constant times $(n_1+n_2)r$.