On the interval of fluctuation of the singular values of random matrices
Olivier GuédonUniversity Paris-Est, Marne-la-Vallée, France
Alexander E. LitvakUniversity of Alberta, Edmonton, Canada
Alain PajorUniversity Paris-Est, Marne-la-Vallée, France
Nicole Tomczak-JaegermannUniversity of Alberta, Edmonton, Canada
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. 1469–1505DOI 10.4171/JEMS/697