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arXiv · 1507.03566

Low-rank Solutions of Linear Matrix Equations via Procrustes Flow

Stephen Tu·Ross Boczar·Max Simchowitz·Mahdi Soltanolkotabi·Benjamin Recht
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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 $(n_1+n_2)r$.

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Stephen Tu, Ross Boczar, Max Simchowitz, Mahdi Soltanolkotabi, Benjamin Recht. 2015-07-13. Low-rank Solutions of Linear Matrix Equations via Procrustes Flow. https://arxiv.org/abs/1507.03566

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