High-Rank Matrix Completion

11 years 7 months ago

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This paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. This generalizes the standard low-rank matrix completion problem to situations in which the matrix rank can be quite high or even full rank. Since the columns belong to a union of subspaces, this problem may also be viewed as a missing-data version of the subspace clustering problem. Let X be an n×N matrix whose (complete) columns lie in a union of at most k subspaces, each of rank ≤ r < n, and assume N ≫ kn. The main result of the paper shows that under mild assumptions each column of X can be perfectly recovered with high probability from an incomplete version so long as at least CrN log2 (n) entries of X are observed uniformly at random, with C > 1 a constant depending on the usual incoherence conditions, the geometrical arrangement of subspaces, and the distribution of columns over the sub...

Brian Eriksson, Laura Balzano, Robert Nowak

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