High-Rank Matrix Completion and Subspace Clustering with Missing Data

TL;DR

This paper introduces a method for completing matrices with many missing entries by leveraging the union of subspace structures, enabling accurate recovery even with high-rank matrices and partial data.

Contribution

It extends low-rank matrix completion to union of subspaces, providing probabilistic guarantees for exact recovery with incomplete data.

Findings

Exact recovery of columns with high probability from partial observations.

Recovery guaranteed when a logarithmic number of entries per column are observed.

Demonstrates practical applications in Internet topology and distance matrix completion.

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 n x 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 CrNlog^2(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 subspaces. The result is illustrated with numerical experiments and an application to Internet distance matrix completion and topology identification.