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Mathematical Notes

, Volume 94, Issue 1–2, pp 107–114 | Cite as

On exact recovery of sparse vectors from linear measurements

  • S. V. Konyagin
  • Yu. V. Malykhin
  • K. S. Ryutin
Article

Abstract

Let 1 ≤ kn < N. We say that a vector x ∈ ℝ N is k-sparse if it has at most k nonzero coordinates. Let Φ be an n × N matrix. We consider the problem of recovery of a k-sparse vector x ∈ ℝ N from the vector y = Φx ∈ ℝ n . We obtain almost-sharp necessary conditions for k, n, N for this problem to be reduced to that of minimization of the ℓ1-norm of vectors z satisfying the condition y = Φz.

Keywords

compressed sensing exact recovery of a k-sparse vector restricted isometry property element of best approximation estimates of Kolmogorov widths 

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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • S. V. Konyagin
    • 1
  • Yu. V. Malykhin
    • 1
  • K. S. Ryutin
    • 2
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Moscow State UniversityMoscowRussia

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