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• W2171046470 abstract "This paper considers the matrix completion problem. We show that it is not necessary to assume joint incoherence, which is a standard but unintuitive and restrictive condition that is imposed by previous studies. This leads to a sample complexity bound that is order-wise optimal with respect to the incoherence parameter (as well as to the rank $r$ and the matrix dimension $n$ up to a log factor). As a consequence, we improve the sample complexity of recovering a semidefinite matrix from $O(nr^{2}log^{2}n)$ to $O(nrlog^{2}n)$, and the highest allowable rank from $Theta(sqrt{n}/log n)$ to $Theta(n/log^{2}n)$. The key step in proof is to obtain new bounds on the $ell_{infty,2}$-norm, defined as the maximum of the row and column norms of a matrix. To illustrate the applicability of our techniques, we discuss extensions to SVD projection, structured matrix completion and semi-supervised clustering, for which we provide order-wise improvements over existing results. Finally, we turn to the closely-related problem of low-rank-plus-sparse matrix decomposition. We show that the joint incoherence condition is unavoidable here for polynomial-time algorithms conditioned on the Planted Clique conjecture. This means it is intractable in general to separate a rank-$omega(sqrt{n})$ positive semidefinite matrix and a sparse matrix. Interestingly, our results show that the standard and joint incoherence conditions are associated respectively with the information (statistical) and computational aspects of the matrix decomposition problem." @default.
• W2171046470 created "2016-06-24" @default.
• W2171046470 creator A5060774236 @default.
• W2171046470 date "2015-05-01" @default.
• W2171046470 modified "2023-09-30" @default.
• W2171046470 title "Incoherence-Optimal Matrix Completion" @default.
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• W2171046470 doi "https://doi.org/10.1109/tit.2015.2415195" @default.
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