Low-rank Solutions of Linear Matrix Equations via Procrustes Flow

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Stephen Tu, Ross Boczar, Max Simchowitz, Mahdi Soltanolkotabi, Ben Recht ;
Proceedings of The 33rd International Conference on Machine Learning, PMLR 48:964-973, 2016.

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 n1 \times n2 matrix of rank r when the number of measurements exceeds a constant times (n1 + n2)r.

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