Recovering low-rank matrices from few coefficients in any basis

TL;DR

This paper introduces simpler, more general techniques for low-rank matrix recovery from few coefficients in any basis, extending matrix completion results with tighter bounds and broad applicability.

Contribution

It presents novel, elementary methods for low-rank matrix recovery that generalize previous approaches and improve bounds, applicable to any basis with incoherence properties.

Findings

Efficient recovery from O(n r nu log^2 n) coefficients in any basis.

Special case of matrix completion with similar bounds.

Tighter bounds for bases incoherent to all low-rank matrices.

Abstract

We present novel techniques for analyzing the problem of low-rank matrix recovery. The methods are both considerably simpler and more general than previous approaches. It is shown that an unknown (n x n) matrix of rank r can be efficiently reconstructed from only O(n r nu log^2 n) randomly sampled expansion coefficients with respect to any given matrix basis. The number nu quantifies the "degree of incoherence" between the unknown matrix and the basis. Existing work concentrated mostly on the problem of "matrix completion" where one aims to recover a low-rank matrix from randomly selected matrix elements. Our result covers this situation as a special case. The proof consists of a series of relatively elementary steps, which stands in contrast to the highly involved methods previously employed to obtain comparable results. In cases where bounds had been known before, our estimates are slightly tighter. We discuss operator bases which are incoherent to all low-rank matrices simultaneously. For these bases, we show that O(n r nu log n) randomly sampled expansion coefficients suffice to recover any low-rank matrix with high probability. The latter bound is tight up to multiplicative constants.