Information-theoretic limits of matrix completion

Authors

Erwin Riegler, David Stotz, and Helmut Bölcskei

Reference

Proc. of IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, pp. 1836-1840, June 2015.

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Abstract

We propose an information-theoretic framework for matrix completion. The theory goes beyond the low-rank structure and applies to general matrices of “low description complexity”. Specifically, we consider m x n random matrices X of arbitrary distribution (continuous, discrete, discrete-continuous mixture, or even singular). With S an ε-support set of X, i.e., P[X in S] ≥ 1 − ε, and dim_B(S) denoting the lower Minkowski dimension of S, we show that k > dim_B(S) trace inner product measurements with measurement matrices A_i, suffice to recover X with probability of error at most ε. The result holds for Lebesgue a.a. A_i and does not need incoherence between the A_i and the unknown matrix X. We furthermore show that k > dim_B(S) measurements also suffice to recover the unknown matrix X from measurements taken with rank-one A_i, again this applies to a.a. rank-one A_i. Rank-one measurement matrices are attractive as they require less storage space than general measurement matrices and can be applied faster. Particularizing our results to the recovery of low-rank matrices, we find that k > (m+n−r)r measurements are sufficient to recover matrices of rank at most r. Finally, we construct a class of rank-r matrices that can be recovered with arbitrarily small probability of error from k < (m + n − r)r measurements.

Keywords

Matrix completion, compressive sensing, Minkowski dimension, information theory

The version posted here corrects a mathematical typo in the version published in the proceedings. Specifically, we added a missing logarithm in the second line from the bottom in the left column on page 5.


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