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Mat. Zametki, 2013, Volume 94, Issue 1, Pages 122–129 (Mi mz10272)  

On Exact Recovery of Sparse Vectors from Linear Measurements

S. V. Konyagin, Yu. V. Malykhin, C. S. Rjutin

M. V. Lomonosov Moscow State University

Abstract: Let $1\le k\le n<N$. We say that a vector $x\in\mathbb R^N$ is $k$-sparse if it has at most $k$ nonzero coordinates. Let $\Phi$ be an $n\times N$ matrix. We consider the problem of recovery of a $k$-sparse vector $x\in\mathbb R^N$ from the vector $y=\Phi x\in\mathbb R^n$. We obtain almost-sharp necessary conditions for $k,n,N$ for this problem to be reduced to that of minimization of the $\ell_1$-norm of vectors $z$ satisfying the condition $y=\Phi z$.

Keywords: compressed sensing, exact recovery of a $k$-sparse vector, restricted isometry property, element of best approximation, estimates of Kolmogorov widths.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00329
Ministry of Education and Science of the Russian Federation НШ-6003.2012.1
НШ-6431.2012.1


DOI: https://doi.org/10.4213/mzm10272

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English version:
Mathematical Notes, 2013, 94:1, 107–114

Bibliographic databases:

Document Type: Article
UDC: 517.518+519.651
Received: 15.11.2012
Revised: 06.05.2013

Citation: S. V. Konyagin, Yu. V. Malykhin, C. S. Rjutin, “On Exact Recovery of Sparse Vectors from Linear Measurements”, Mat. Zametki, 94:1 (2013), 122–129; Math. Notes, 94:1 (2013), 107–114

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    Remarks
    • Letter to the Editor
      S. V. Konyagin, Yu. V. Malykhin, K. S. Ryutin
      Mat. Zametki, 2014, 96:3, 480
  • Математические заметки Mathematical Notes
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