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 $δ_{k}^A+θ_{k,k}^A < 1$ 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 $ε> 0$, the condition $δ_k^A + θ_{k, k}^A < 1+ε$ is not sufficient to guarantee such exact recovery using any recovery method. Similarly, for affine rank minimization, if $δ_{r}^\mathcal{M}+θ_{r,r}^\mathcal{M}< 1$ 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 $ε> 0$, $δ_r^\mathcal{M} +θ_{r,r}^\mathcal{M} < 1+ε$ does not ensure such exact recovery using any method. Moreover, in the noisy case the conditions $δ_{k}^A+θ_{k,k}^A < 1$ and $δ_{r}^\mathcal{M}+θ_{r,r}^\mathcal{M}< 1$ are also sufficient for the stable recovery of sparse signals and low-rank matrices respectively. Applications and extensions are also discussed.

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