Obtaining error-minimizing estimates and universal entry-wise error bounds for low-rank matrix completion

TL;DR

This paper introduces a comprehensive framework for reconstructing and denoising individual entries in incomplete, noisy low-rank matrices, providing algorithms and error bounds that improve upon existing methods.

Contribution

It presents a novel, general approach for entry-wise reconstruction and error estimation, including a fast algorithm for rank-one matrices with near-optimal error minimization.

Findings

Exact reconstruction in noiseless cases.

Fast, parallelizable algorithm for rank-one matrices.

Error bounds close to state-of-the-art methods.

Abstract

We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individually. In the noiseless case our algorithm is exact. For rank-one matrices, the new algorithm is fast, admits a highly-parallel implementation, and produces an error minimizing estimate that is qualitatively close to our theoretical and the state-of-the-are Nuclear Norm and OptSpace methods.