Smoothed analysis in compressed sensing
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909707870928896 |
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| author | Aigner-Horev, Elad Hefetz, Dan Trushkin, Michael |
| author_facet | Aigner-Horev, Elad Hefetz, Dan Trushkin, Michael |
| contents | Arbitrary matrices $M \in \mathbb{R}^{m \times n}$, randomly perturbed in an additive manner using a random matrix $R \in \mathbb{R}^{m \times n}$, are shown to asymptotically almost surely satisfy the so-called {\sl robust null space property}. Whilst insisting on an asymptotically optimal order of magnitude for $m$ required to attain {\sl unique reconstruction} via $\ell_1$-minimisation algorithms, our results track the level of arbitrariness allowed for the fixed seed matrix $M$ as well as the degree of distributional irregularity allowed for the entries of the perturbing matrix $R$. Starting with sub-gaussian entries for $R$, our results culminate with these allowed to have substantially heavier tails than sub-exponential ones. Throughout this trajectory, two measures control the arbitrariness allowed for $M$; the first is $\|M\|_\infty$ and the second is a localised notion of the Frobenius norm of $M$ (which depends on the sparsity of the signal being reconstructed). A key tool driving our proofs is {\sl Mendelson's small-ball method} ({\em Learning without concentration}, J. ACM, Vol. $62$, $2015$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05188 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smoothed analysis in compressed sensing Aigner-Horev, Elad Hefetz, Dan Trushkin, Michael Probability Information Theory Arbitrary matrices $M \in \mathbb{R}^{m \times n}$, randomly perturbed in an additive manner using a random matrix $R \in \mathbb{R}^{m \times n}$, are shown to asymptotically almost surely satisfy the so-called {\sl robust null space property}. Whilst insisting on an asymptotically optimal order of magnitude for $m$ required to attain {\sl unique reconstruction} via $\ell_1$-minimisation algorithms, our results track the level of arbitrariness allowed for the fixed seed matrix $M$ as well as the degree of distributional irregularity allowed for the entries of the perturbing matrix $R$. Starting with sub-gaussian entries for $R$, our results culminate with these allowed to have substantially heavier tails than sub-exponential ones. Throughout this trajectory, two measures control the arbitrariness allowed for $M$; the first is $\|M\|_\infty$ and the second is a localised notion of the Frobenius norm of $M$ (which depends on the sparsity of the signal being reconstructed). A key tool driving our proofs is {\sl Mendelson's small-ball method} ({\em Learning without concentration}, J. ACM, Vol. $62$, $2015$). |
| title | Smoothed analysis in compressed sensing |
| topic | Probability Information Theory |
| url | https://arxiv.org/abs/2505.05188 |