Completing Low-Rank Matrices with Corrupted Samples from Few Coefficients in General Basis

TL;DR

This paper demonstrates that the range space of a low-rank matrix can be exactly recovered from few coefficients in a general basis, extending robust matrix completion to higher ranks and more general settings.

Contribution

It proves exact recovery from few coefficients in general basis, introduces a universal regularization parameter, and proposes a computationally efficient filtering algorithm.

Findings

Exact recovery possible with high rank and corrupted samples.

Universal regularization parameter $\\lambda=1/\\sqrt{\log n}$ is effective.

Theoretical guarantees for the proposed filtering algorithm.

Abstract

Subspace recovery from corrupted and missing data is crucial for various applications in signal processing and information theory. To complete missing values and detect column corruptions, existing robust Matrix Completion (MC) methods mostly concentrate on recovering a low-rank matrix from few corrupted coefficients w.r.t. standard basis, which, however, does not apply to more general basis, e.g., Fourier basis. In this paper, we prove that the range space of an $m\times n$ matrix with rank $r$ can be exactly recovered from few coefficients w.r.t. general basis, though $r$ and the number of corrupted samples are both as high as $O(\min\{m,n\}/\log^3 (m+n))$. Our model covers previous ones as special cases, and robust MC can recover the intrinsic matrix with a higher rank. Moreover, we suggest a universal choice of the regularization parameter, which is $\lambda=1/\sqrt{\log n}$. By our $\ell_{2,1}$ filtering algorithm, which has theoretical guarantees, we can further reduce the computational cost of our model. As an application, we also find that the solutions to extended robust Low-Rank Representation and to our extended robust MC are mutually expressible, so both our theory and algorithm can be applied to the subspace clustering problem with missing values under certain conditions. Experiments verify our theories.