Matrix Completion with Noise via Leveraged Sampling
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Many matrix completion methods assume that the data follows the uniform distribution. To address the limitation of this assumption, Chen et al. <cit.> propose to recover the matrix where the data follows the specific biased distribution. Unfortunately, in most real-world applications, the recovery of a data matrix appears to be incomplete, and perhaps even corrupted information. This paper considers the recovery of a low-rank matrix, where some observed entries are sampled in a biased distribution suitably dependent on leverage scores of a matrix, and some observed entries are uniformly corrupted. Our theoretical findings show that we can provably recover an unknown n× n matrix of rank r from just about O(nrlog^2 n) entries even when the few observed entries are corrupted with a small amount of noisy information. Empirical studies verify our theoretical results.
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