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. Pajor
  • N. Tomczak-Jaegermann
Article

Abstract

This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X1,…,±XN∈ℝn, (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⊂ℝn is a convex body and X1,…,XNK 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).

Keywords

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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References

  1. 1.
    Adamczak, R., Litvak, A.E., Pajor, A., Tomczak-Jaegermann, N.: Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles. J. Am. Math. Soc. 234, 535–561 (2010) MathSciNetGoogle Scholar
  2. 2.
    Baraniuk, R., Davenport, M., DeVore, R., Wakin, M.: A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28, 253–263 (2008) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Barthe, F., Guédon, O., Mendelson, S., Naor, A.: A probabilistic approach to the geometry of the lpn ball. Ann. Probab. 33, 480–513 (2005) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Borell, C.: The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30, 207–216 (1975) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Candes, E.: The restricted isometry property and its implications for compressed sensing. C. R. Acad. Sci. Paris, Ser. I 346, 589–592 (2008) MathSciNetMATHGoogle Scholar
  6. 6.
    Candes, E., Romberg, J., Tao, T.: Stable signal recovery from incomplete and incurable measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Candes, E., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Candes, E., Tao, T.: Near-optimal signal recovery from random projections: universal encoding strategies. IEEE Trans. Inf. Theory 52, 5406–5425 (2006) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cohen, A., Dahmen, W., Devore, R.: Compressed sensing and k-term approximation. J. Am. Math. Soc. 22, 211–231 (2009) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Donoho, D.L.: Neighborly polytopes and sparse solutions of underdetermined linear equations, Department of Statistics, Stanford University (2005) Google Scholar
  11. 11.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52, 1289–1306 (2006) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Donoho, D.L.: High-dimensional centrally-symmetric polytopes with neighborliness proportional to dimension. Discrete Comput. Geom. 35, 617–652 (2006) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Donoho, D.L., Tanner, J.: Counting faces of randomly projected polytopes when the projection radically lowers dimension. J. Am. Math. Soc. 22, 1–53 (2009) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Fleury, B., Guédon, O., Paouris, G.: A stability result for mean width of L p-centroid bodies. Adv. Math. 214, 865–877 (2007) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Grünbaum, B.: Convex Polytopes, 2nd edn. Graduate Texts in Mathematics, vol. 221. Springer, New York (2003) Google Scholar
  16. 16.
    Hitczenko, P., Montgomery-Smith, S.J., Oleszkiewicz, K.: Moment inequalities for sums of certain independent symmetric random variables. Stud. Math. 123, 15–42 (1997) MathSciNetMATHGoogle Scholar
  17. 17.
    Kashin, B.S., Temlyakov, V.N.: A remark on compressed sensing. Math. Not. 82, 748–755 (2007) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Klartag, B.: Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245, 284–310 (2007) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Klartag, B.: A central limit theorem for convex sets. Invent. Math. 168, 91–131 (2007) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Latała, R.: Tail and moment estimates for sums of independent random vectors with logarithmically concave tails. Stud. Math. 118, 301–304 (1996) MATHGoogle Scholar
  21. 21.
    Linial, N., Novik, I.: How neighborly can a centrally symmetric polytope be? Discrete Comput. Geom. 36, 273–281 (2006) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Mankiewicz, P., Tomczak-Jaegermann, N.: Stability properties of neighborly random polytopes. Discrete Comput. Geom. 41, 257–272 (2009) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Mendelson, S., Pajor, A., Tomczak-Jaegermann, N.: Reconstruction and sub-gaussian processes. C. R. Acad. Sci. Paris 340, 885–888 (2005) MathSciNetMATHGoogle Scholar
  24. 24.
    Mendelson, S., Pajor, A., Tomczak-Jaegermann, N.: Reconstruction and sub-gaussian operators. Geom. Funct. Anal. 17, 1248–1282 (2007) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Mendelson, S., Pajor, A., Tomczak-Jaegermann, N.: Uniform uncertainty principle for Bernoulli and sub-gaussian ensembles. Constr. Approx. 28, 277–289 (2008) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Paouris, G.: Concentration of mass on convex bodies. Geom. Funct. Anal. 16, 1021–1049 (2006) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Rudelson, M., Vershynin, R.: Geometric approach to error correcting codes and reconstruction of signals. Int. Math. Res. Not. 64, 4019–4041 (2005) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Talagrand, M.: The Generic Chaining. Upper and Lower Bounds of Stochastic Processes. Springer Monographs in Mathematics. Springer, Berlin (2005) MATHGoogle Scholar
  29. 29.
    Talagrand, M.: The supremum of some canonical processes. Am. J. Math. 116, 283–325 (1994) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. With Applications to Statistics. Springer Series in Statistics. Springer, New York (1996) MATHGoogle Scholar
  31. 31.
    Ziegler, G.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. Adamczak
    • 1
  • A. E. Litvak
    • 2
  • A. Pajor
    • 3
  • 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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