Rank minimization via online learning
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Large-Scale Parallel Collaborative Filtering for the Netflix Prize
AAIM '08 Proceedings of the 4th international conference on Algorithmic Aspects in Information and Management
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IEEE Transactions on Information Theory
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SIAM Journal on Optimization
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Journal of the ACM (JACM)
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Low rank modeling of signed networks
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
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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 [17]. In the alternating minimization approach, the low-rank target matrix is written in a bi-linear form, i.e. X = UV†; the algorithm then alternates between finding the best U and the best V. Typically, each alternating step in isolation is convex and tractable. However the overall problem becomes non-convex and is prone to local minima. In fact, there has been almost no theoretical understanding of when this approach yields a good result. In this paper we present one of the first theoretical analyses 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 significantly simpler analysis.