The Stability of Low-Rank Matrix Reconstruction: a Constrained Singular Value View

TL;DR

This paper introduces the $ ext{l}_*$-constrained minimal singular value ($ ext{l}_*$-CMSV) as a new measure to analyze the stability of low-rank matrix reconstruction, offering more concise performance bounds and practical insights.

Contribution

It proposes the $ ext{l}_*$-CMSV as a novel, less complex stability measure for low-rank matrix recovery, with theoretical bounds and analysis for various measurement operators.

Findings

$ ext{l}_*$-CMSV bounds are high probability for isotropic/subgaussian operators

Performance bounds using $ ext{l}_*$-CMSV are more concise than previous methods

Fixed point characterization of $ ext{l}_*$-CMSV aids in computation

Abstract

The stability of low-rank matrix reconstruction with respect to noise is investigated in this paper. The $\ell_*$-constrained minimal singular value ($\ell_*$-CMSV) of the measurement operator is shown to determine the recovery performance of nuclear norm minimization based algorithms. Compared with the stability results using the matrix restricted isometry constant, the performance bounds established using $\ell_*$-CMSV are more concise, and their derivations are less complex. Isotropic and subgaussian measurement operators are shown to have $\ell_*$-CMSVs bounded away from zero with high probability, as long as the number of measurements is relatively large. The $\ell_*$-CMSV for correlated Gaussian operators are also analyzed and used to illustrate the advantage of $\ell_*$-CMSV compared with the matrix restricted isometry constant. We also provide a fixed point characterization of $\ell_*$-CMSV that is potentially useful for its computation.