Constructive Approximation

, Volume 34, Issue 1, pp 61–88 | Cite as

Restricted Isometry Property of Matrices with Independent Columns and Neighborly Polytopes by Random Sampling

  • R. Adamczak
  • A. E. Litvak
  • A. PajorEmail author
  • N. Tomczak-Jaegermann


This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X 1,…,±X N ∈ℝ n , (Nn). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property with overwhelming probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a subexponential tail inequality possess the restricted isometry property with overwhelming probability. We show that such “sensing” matrices are valid for the exact reconstruction process of m-sparse vectors via 1 minimization with mCn/log 2(cN/n). The class of sampling matrices we study includes the case of matrices with columns that are independent isotropic vectors with log-concave densities. We deduce that if K⊂ℝ n is a convex body and X 1,…,X N K are i.i.d. random vectors uniformly distributed on K, then, with overwhelming probability, the symmetric convex hull of these points is an m-centrally-neighborly polytope with mn/log 2(cN/n).


Centrally-neighborly polytopes Compressed sensing Random matrices Restricted isometry property Underdetermined systems of linear equations 

Mathematics Subject Classification (2000)

52A23 60B20 94A12 52B12 46B06 15B52 41A45 94B75 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. Adamczak
    • 1
  • A. E. Litvak
    • 2
  • A. Pajor
    • 3
    Email author
  • N. Tomczak-Jaegermann
    • 2
  1. 1.Institute of MathematicsUniversity of WarsawWarszawaPoland
  2. 2.Dept. of Math. and Stat. SciencesUniversity of AlbertaEdmontonCanada
  3. 3.Équipe d’Analyse et Mathématiques AppliquéesUniversité Paris-EstMarne-la-Vallée, Cedex 2France

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