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# High-Rank Matrix Completion

*Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics*, PMLR 22:373-381, 2012.

#### Abstract

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 nxN matrix whose (complete) columns lie in a union of at most k subspaces, each of rank = r n, and assume Nkn. 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 C r N \log^2(n) entries of X are observed uniformly at random, with C1 a constant depending on the usual incoherence conditions, the geometrical arrangement of subspaces, and the distribution of columns over the subspaces. The result is illustrated with numerical experiments and an application to Internet distance matrix completion and topology identification.