Guaranteed Rank Minimization via Singular Value Projection

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 under affine constraints ARMP and show that SVP recovers the minimum rank solution for affine constraints that satisfy a Restricted Isometry Property} (RIP). Our method guarantees geometric convergence rate even in the presence of noise and requires strictly weaker assumptions on the RIP constants than the existing methods. We also introduce a Newton-step for our SVP framework to speed-up the convergence with substantial empirical gains. Next, we address a practically important application of ARMP - the problem of low-rank matrix completion, for which the defining affine constraints do not directly obey RIP, hence the guarantees of SVP do not hold. However, we provide partial progress towards a proof of exact recovery for our algorithm by showing a more restricted isometry property and observe empirically that our algorithm recovers low-rank Incoherent matrices from an almost optimal number of uniformly sampled entries. We also demonstrate empirically that our algorithms outperform existing methods, such as those of \cite{CaiCS2008,LeeB2009b, KeshavanOM2009}, for ARMP and the matrix completion problem by an order of magnitude and are also more robust to noise and sampling schemes. In particular, results show that our SVP-Newton method is significantly robust to noise and performs impressively on a more realistic power-law sampling scheme for the matrix completion problem.