Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices
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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>4/3, in compressed sensing δ_tk^A < √((t-1)/t) guarantees the exact recovery of all k sparse signals in the noiseless case through the constrained ℓ_1 minimization, and similarly in affine rank minimization δ_tr^M< √((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 ϵ>0, δ_tk^A<√(t-1/t)+ϵ 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 δ_tk^A < √((t-1)/t) and δ_tr^M< √((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.
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