Compressed Sensing and Affine Rank Minimization under Restricted Isometry

TL;DR

This paper introduces new restricted isometry conditions that guarantee exact and stable recovery of sparse signals and low-rank matrices in compressed sensing and affine rank minimization, establishing sharp bounds for these guarantees.

Contribution

The paper derives precise restricted isometry conditions involving $ heta$-constants that are both necessary and sufficient for exact and stable recovery in compressed sensing and affine rank minimization.

Findings

Conditions $ ext{delta}_k^A + heta_{k,k}^A < 1$ guarantee exact recovery.

Conditions $ ext{delta}_r^ ext{M} + heta_{r,r}^ ext{M} < 1$ guarantee exact recovery.

The bounds are sharp; exceeding them invalidates guarantees.

Abstract

This paper establishes new restricted isometry conditions for compressed sensing and affine rank minimization. It is shown for compressed sensing that $\delta_{k}^A+\theta_{k,k}^A < 1$ guarantees the exact recovery of all $k$ sparse signals in the noiseless case through the constrained $\ell_1$ minimization. Furthermore, the upper bound 1 is sharp in the sense that for any $\epsilon > 0$, the condition $\delta_k^A + \theta_{k, k}^A < 1+\epsilon$ is not sufficient to guarantee such exact recovery using any recovery method. Similarly, for affine rank minimization, if $\delta_{r}^\mathcal{M}+\theta_{r,r}^\mathcal{M}< 1$ then all matrices with rank at most $r$ can be reconstructed exactly in the noiseless case via the constrained nuclear norm minimization; and for any $\epsilon > 0$, $\delta_r^\mathcal{M} +\theta_{r,r}^\mathcal{M} < 1+\epsilon$ does not ensure such exact recovery using any method. Moreover, in the noisy case the conditions $\delta_{k}^A+\theta_{k,k}^A < 1$ and $\delta_{r}^\mathcal{M}+\theta_{r,r}^\mathcal{M}< 1$ are also sufficient for the stable recovery of sparse signals and low-rank matrices respectively. Applications and extensions are also discussed.