On the interval of fluctuation of the singular values of random matrices

  • Olivier Guédon

    University Paris-Est, Marne-la-Vallée, France
  • Alexander E. Litvak

    University of Alberta, Edmonton, Canada
  • Alain Pajor

    University Paris-Est, Marne-la-Vallée, France
  • Nicole Tomczak-Jaegermann

    University of Alberta, Edmonton, Canada

Abstract

Let be a matrix whose columns are independent random vectors in . Assume that the tails of the 1-dimensional marginals decay as uniformly in and . Then for we prove that with high probability has the Restricted Isometry Property (RIP) provided that Euclidean norms are concentrated around . We also show that the covariance matrix is well approximated by empirical covariance matrices and establish corresponding quantitative estimates on the rate of convergence in terms of the ratio . Moreover, we obtain sharp bounds for both problems when the decay is of the type exp , with , extending the known case .

Cite this article

Olivier Guédon, Alexander E. Litvak, Alain Pajor, Nicole Tomczak-Jaegermann, On the interval of fluctuation of the singular values of random matrices. J. Eur. Math. Soc. 19 (2017), no. 5, pp. 1469–1505

DOI 10.4171/JEMS/697