Compressed Sensing and Affine Rank Minimization under Restricted Isometry

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 $\ell1$ minimization. Furthermore, the upper bound 1 is sharp in the sense that for any $\epsilon > 0$, the condition $\deltak^A + \theta{k, k}^A < 1+\epsilon$ 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 $\epsilon > 0$, $\deltar^\mathcal{M} +\theta{r,r}^\mathcal{M} < 1+\epsilon$ 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.