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• W2953302126 abstract "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 $tge {4/3}$, 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." @default.
• W2953302126 created "2019-06-27" @default.
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• W2953302126 date "2013-06-05" @default.
• W2953302126 modified "2023-09-26" @default.
• W2953302126 title "Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices" @default.
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• W2953302126 doi "https://doi.org/10.48550/arxiv.1306.1154" @default.
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