Performance Guarantees for Schatten-$p$ Quasi-Norm Minimization in Recovery of Low-Rank Matrices

TL;DR

This paper provides theoretical guarantees for Schatten-$p$ quasi-norm minimization in low-rank matrix recovery, extending RIP-based conditions and ensuring exact recovery under certain null space properties.

Contribution

It introduces new null space and RIP-based sufficient conditions for exact low-rank matrix recovery using Schatten-$p$ quasi-norm minimization, generalizing previous nuclear norm results.

Findings

Null space property guarantees exact recovery of certain low-rank matrices.

RIP-based conditions for Schatten-$p$ minimization are established.

Theoretical extension of $ ext{RIP}$ conditions from $ ext{l}_p$ to Schatten-$p$ quasi-norms.

Abstract

We address some theoretical guarantees for Schatten-$p$ quasi-norm minimization ($p \in (0,1]$) in recovering low-rank matrices from compressed linear measurements. Firstly, using null space properties of the measurement operator, we provide a sufficient condition for exact recovery of low-rank matrices. This condition guarantees unique recovery of matrices of ranks equal or larger than what is guaranteed by nuclear norm minimization. Secondly, this sufficient condition leads to a theorem proving that all restricted isometry property (RIP) based sufficient conditions for $\ell_p$ quasi-norm minimization generalize to Schatten-$p$ quasi-norm minimization. Based on this theorem, we provide a few RIP-based recovery conditions.