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

This article is cited in 1 scientific paper (total in 2 paper)

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

Full text: PDF file (500 kB)
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English version:
Mathematical Notes, 2013, 94:1, 107–114

Bibliographic databases:

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

Citation in format AMSBIB
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\paper On Exact Recovery of Sparse Vectors from Linear Measurements
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\issue 1
\pages 122--129
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    Citing articles on Google Scholar: Russian citations, English citations
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    Remarks
    • Letter to the Editor
      S. V. Konyagin, Yu. V. Malykhin, K. S. Ryutin
      Mat. Zametki, 2014, 96:3, 480


    This publication is cited in the following articles:
    1. S. V. Konyagin, Yu. V. Malykhin, K. S. Ryutin, “Letter to the Editor”, Math. Notes, 96:3 (2014), 462–462  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. A. A. Vasileva, “Kolmogorovskie poperechniki peresecheniya dvukh konechnomernykh sharov”, Izv. vuzov. Matem., 2021, no. 7, 23–29  mathnet  crossref
  • Математические заметки Mathematical Notes
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