This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while leads to sharp results. It is shown that for any given constant $t\ge { 4/3}$, in compressed sensing $\delta_{ tk}^A < \sqrt{ (t-1)/t}$ guarantees the exact recovery of all $k$ sparse signals in the noiseless case through the constrained $\ell_1$ minimization, and similarly in affine rank minimization $\delta_{ tr}^\mathcal{ M}< \sqrt{ (t-1)/t}$ ensures the exact reconstruction of all matrices with rank at most $r$ in the noiseless case via the constrained nuclear norm minimization. Moreover, for any $\epsilon>0$, $\delta_{ tk}^A<\sqrt{ \frac{ t-1}{ t}}+\epsilon$ is not sufficient to guarantee the exact recovery of all $k$-sparse signals for large $k$. Similar result also holds for matrix recovery. In addition, the conditions $\delta_{ tk}^A < \sqrt{ (t-1)/t}$ and $\delta_{ tr}^\mathcal{ M}< \sqrt{ (t-1)/t}$ are also shown to be sufficient respectively for stable recovery of approximately sparse signals and low-rank matrices in the noisy case.
Categories: cs.IT math.IT math.ST stat.ML stat.TH