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

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Abstract

This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X 1,…,±X N ∈ℝn, (N≥n). 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 ℓ 1 minimization with m≤Cn/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⊂ℝn 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 m∼n/log 2(cN/n).

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Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland

    R. Adamczak

  2. Dept. of Math. and Stat. Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1

    A. E. Litvak & N. Tomczak-Jaegermann

  3. Équipe d’Analyse et Mathématiques Appliquées, Université Paris-Est, 5, boulevard Descartes, Champs sur Marne, 77454, Marne-la-Vallée, Cedex 2, France

    A. Pajor

Authors
  1. R. Adamczak
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  2. A. E. Litvak
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  3. A. Pajor
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  4. N. Tomczak-Jaegermann
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Corresponding author

Correspondence to A. Pajor.

Additional information

Communicated by Emmanuel J. Candes.

The research of R. Adamczak was partially supported by the Foundation for Polish Science.

N. Tomczak-Jaegermann holds the Canada Research Chair in Geometric Analysis.

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Adamczak, R., Litvak, A.E., Pajor, A. et al. Restricted Isometry Property of Matrices with Independent Columns and Neighborly Polytopes by Random Sampling. Constr Approx 34, 61–88 (2011). https://doi.org/10.1007/s00365-010-9117-4

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  • DOI: https://doi.org/10.1007/s00365-010-9117-4

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