Restricted strong convexity and weighted matrix completion: Optimal bounds with noise
Abstract
We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong convexity with respect to weighted Frobenius norm. Using this property, we obtain as corollaries a number of error bounds on matrix completion in the weighted Frobenius norm under noisy sampling and for both exact and near low-rank matrices. Our results are based on measures of the "spikiness" and "low-rankness" of matrices that are less restrictive than the incoherence conditions imposed in previous work. Our technique involves an $M$-estimator that includes controls on both the rank and spikiness of the solution, and we establish non-asymptotic error bounds in weighted Frobenius norm for recovering matrices lying with $\ell_q$-"balls" of bounded spikiness. Using information-theoretic methods, we show that no algorithm can achieve better estimates (up to a logarithmic factor) over these same sets, showing that our conditions on matrices and associated rates are essentially optimal.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2010
- DOI:
- 10.48550/arXiv.1009.2118
- arXiv:
- arXiv:1009.2118
- Bibcode:
- 2010arXiv1009.2118N
- Keywords:
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- Computer Science - Information Theory;
- Mathematics - Statistics Theory