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In this paper we study the problem of recovering a low-rank matrix fromlinear measurements. Our algorithm, which we call Procrustes Flow, starts froman initial estimate obtained by a thresholding scheme followed by gradientdescent on a non-convex objective. We show that as long as the measurementsobey a standard restricted isometry property, our algorithm converges to theunknown matrix at a geometric rate. In the case of Gaussian measurements, suchconvergence occurs for a $n_1 \times n_2$ matrix of rank $r$ when the number ofmeasurements exceeds a constant times $(n_1+n_2)r$.

Low-rank Solutions of Linear Matrix Equations via Procrustes Flow