A Simplified Approach to Recovery Conditions for Low Rank Matrices

TL;DR

This paper simplifies the derivation of recovery conditions for low-rank matrices, extending known vector results to matrices using a key singular value inequality, thus providing clearer and improved theoretical guarantees.

Contribution

It introduces a straightforward method to extend recovery conditions from vectors to matrices, achieving the best known restricted isometry and nullspace conditions for matrix recovery.

Findings

Extended robust recovery conditions from vectors to matrices.

Achieved the best known restricted isometry conditions for matrices.

Provided a simple, transparent proof technique using singular value inequalities.

Abstract

Recovering sparse vectors and low-rank matrices from noisy linear measurements has been the focus of much recent research. Various reconstruction algorithms have been studied, including $\ell_1$ and nuclear norm minimization as well as $\ell_p$ minimization with $p<1$. These algorithms are known to succeed if certain conditions on the measurement map are satisfied. Proofs of robust recovery for matrices have so far been much more involved than in the vector case. In this paper, we show how several robust classes of recovery conditions can be extended from vectors to matrices in a simple and transparent way, leading to the best known restricted isometry and nullspace conditions for matrix recovery. Our results rely on the ability to "vectorize" matrices through the use of a key singular value inequality.