Completing Any Lowrank Matrix, Provably
Abstract
Matrix completion, i.e., the exact and provable recovery of a lowrank matrix from a small subset of its elements, is currently only known to be possible if the matrix satisfies a restrictive structural constraintknown as {\em incoherence}on its row and column spaces. In these cases, the subset of elements is sampled uniformly at random. In this paper, we show that {\em any} rank$ r $ $ n$by$ n $ matrix can be exactly recovered from as few as $O(nr \log^2 n)$ randomly chosen elements, provided this random choice is made according to a {\em specific biased distribution}: the probability of any element being sampled should be proportional to the sum of the leverage scores of the corresponding row, and column. Perhaps equally important, we show that this specific form of sampling is nearly necessary, in a natural precise sense; this implies that other perhaps more intuitive sampling schemes fail. We further establish three ways to use the above result for the setting when leverage scores are not known \textit{a priori}: (a) a sampling strategy for the case when only one of the row or column spaces are incoherent, (b) a twophase sampling procedure for general matrices that first samples to estimate leverage scores followed by sampling for exact recovery, and (c) an analysis showing the advantages of weighted nuclear/tracenorm minimization over the vanilla unweighted formulation for the case of nonuniform sampling.
 Publication:

arXiv eprints
 Pub Date:
 June 2013
 arXiv:
 arXiv:1306.2979
 Bibcode:
 2013arXiv1306.2979C
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Information Theory;
 Computer Science  Machine Learning
 EPrint:
 Added a new necessary condition(Theorem 6) and a result on completion of row coherent matrices(Corollary 4). Partial results appeared in the International Conference on Machine Learning 2014, under the title 'Coherent Matrix Completion'. (34 pages, 4 figures)