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This paper considers compressed sensing and affine rank minimization in bothnoiseless and noisy cases and establishes sharp restricted isometry conditionsfor sparse signal and low-rank matrix recovery. The analysis relies on a keytechnical tool which represents points in a polytope by convex combinations ofsparse 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 ofall matrices with rank at most $r$ in the noiseless case via the constrainednuclear norm minimization. Moreover, for any $\epsilon>0$,$\delta_{tk}^A<\sqrt{\frac{t-1}{t}}+\epsilon$ is not sufficient to guaranteethe exact recovery of all $k$-sparse signals for large $k$. Similar result alsoholds 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 tobe sufficient respectively for stable recovery of approximately sparse signalsand low-rank matrices in the noisy case.

Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices