IncoherenceOptimal Matrix Completion
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 orderwise 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 $O(nr\log^{2}n)$, 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 semisupervised clustering, for which we provide orderwise improvements over existing results. Finally, we turn to the closelyrelated problem of lowrankplussparse matrix decomposition. We show that the joint incoherence condition is unavoidable here for polynomialtime 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2013
 arXiv:
 arXiv:1310.0154
 Bibcode:
 2013arXiv1310.0154C
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Machine Learning;
 Statistics  Machine Learning
 EPrint:
 Fixed a minor error in Theorem 3 for matrix decomposition. To appear in the IEEE Transactions on Information Theory