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Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization with affine constraints (ARMP) and show that SVP recovers the minimum rank solution for affine constraints that satisfy the restricted isometry property. We show robustness of our method to noise with a strong geometric convergence rate even for noisy measurements. Our results improve upon a recent breakthrough by Recht, Fazel and Parillo [RFP07] in three significant ways: 1) our method (SVP) is significantly simpler to analyse and easier to implement, 2) we give geometric convergence guarantees for SVP and, as demonstrated empiricially, SVP is significantly faster on real-world and synthetic problems, 3) we give optimality and geometric convergence guarantees even for the noisy version of ARMP. In addition, we address the practically important problem of low-rank matrix completion, which can be seen as a special case of ARMP. However, the affine constraints defining the matrix-completion problem do not obey the restricted isometry property in general. We empirically demonstrate that our algorithm recovers low-rank incoherent matrices from an almost optimal number of uniformly sampled entries. We make partial progress towards proving exact recovery and provide some intuition for the performance of SVP applied to matrix completion by showing a more restricted isometry property. Our algorithm outperforms existing methods, such as those of [RFP07, CR08, CT09, CCS08, KOM09], for ARMP and the matrix-completion problem by an order of magnitude and is also significantly more robust to noise.
Guaranteed Rank Minimization via Singular Value Projection