Compressed Sensing and Affine Rank Minimization under Restricted Isometry

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.