33,599 research outputs found
Low-rank Matrix Completion using Alternating Minimization
Alternating minimization represents a widely applicable and empirically
successful approach for finding low-rank matrices that best fit the given data.
For example, for the problem of low-rank matrix completion, this method is
believed to be one of the most accurate and efficient, and formed a major
component of the winning entry in the Netflix Challenge.
In the alternating minimization approach, the low-rank target matrix is
written in a bi-linear form, i.e. ; the algorithm then alternates
between finding the best and the best . Typically, each alternating step
in isolation is convex and tractable. However the overall problem becomes
non-convex and there has been almost no theoretical understanding of when this
approach yields a good result.
In this paper we present first theoretical analysis of the performance of
alternating minimization for matrix completion, and the related problem of
matrix sensing. For both these problems, celebrated recent results have shown
that they become well-posed and tractable once certain (now standard)
conditions are imposed on the problem. We show that alternating minimization
also succeeds under similar conditions. Moreover, compared to existing results,
our paper shows that alternating minimization guarantees faster (in particular,
geometric) convergence to the true matrix, while allowing a simpler analysis
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 with affine constraints (ARMP) and show
that SVP recovers the minimum rank solution for affine constraints that satisfy
the "restricted isometry property" and show robustness of our method to noise.
Our results improve upon a recent breakthrough by Recht, Fazel and Parillo
(RFP07) and Lee and Bresler (LB09) in three significant ways:
1) our method (SVP) is significantly simpler to analyze and easier to
implement,
2) we give recovery guarantees under strictly weaker isometry assumptions
3) we give geometric convergence guarantees for SVP even in presense of noise
and, as demonstrated empirically, SVP is significantly faster on real-world and
synthetic problems.
In addition, we address the practically important problem of low-rank matrix
completion (MCP), which can be seen as a special case of ARMP. 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 strong
performance of SVP applied to matrix completion by showing a more restricted
isometry property. Our algorithm outperforms existing methods, such as those of
\cite{RFP07,CR08,CT09,CCS08,KOM09,LB09}, for ARMP and the matrix-completion
problem by an order of magnitude and is also significantly more robust to
noise.Comment: An earlier version of this paper was submitted to NIPS-2009 on June
5, 200
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