New Null Space Results and Recovery Thresholds for Matrix Rank Minimization

TL;DR

This paper improves null space-based recovery thresholds for nuclear norm minimization in matrix rank minimization, aligning theoretical results more closely with empirical observations, especially for low-rank matrices.

Contribution

It applies recent compressed sensing analysis to NNM null space conditions, significantly enhancing recovery thresholds and providing new insights for positive semidefinite matrices.

Findings

Improved weak recovery thresholds matching simulations

Only three times oversampling needed for linearly growing rank

Enhanced null space conditions for positive semidefinite matrices

Abstract

Nuclear norm minimization (NNM) has recently gained significant attention for its use in rank minimization problems. Similar to compressed sensing, using null space characterizations, recovery thresholds for NNM have been studied in \cite{arxiv,Recht_Xu_Hassibi}. However simulations show that the thresholds are far from optimal, especially in the low rank region. In this paper we apply the recent analysis of Stojnic for compressed sensing \cite{mihailo} to the null space conditions of NNM. The resulting thresholds are significantly better and in particular our weak threshold appears to match with simulation results. Further our curves suggest for any rank growing linearly with matrix size $n$ we need only three times of oversampling (the model complexity) for weak recovery. Similar to \cite{arxiv} we analyze the conditions for weak, sectional and strong thresholds. Additionally a separate analysis is given for special case of positive semidefinite matrices. We conclude by discussing simulation results and future research directions.