Geodesic
On the interval of fluctuation of the singular values of random matrices
Journal of the European Mathematical Society, Tome 19 (2017) no. 5, pp. 1469-1505 Cet article a éte moissonné depuis la source EMS Press

Voir la notice de l'article

Let A be a matrix whose columns X1​,...,XN​ are independent random vectors in Rn. Assume that the tails of the 1-dimensional marginals decay as P(∣〈Xi​,a〉∣≥t)≤Ct−p uniformly in a∈Sn−1 and i≤N. Then for p>4 we prove that with high probability A/n​ has the Restricted Isometry Property (RIP) provided that Euclidean norms ∣Xi​∣ are concentrated around n​. 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 n/N. Moreover, we obtain sharp bounds for both problems when the decay is of the type exp (−tα), with α∈(0,2], extending the known case α∈(1,2].
DOI : 10.4171/jems/697
Classification : 60-XX, 15-XX, 46-XX
Keywords: Random matrices, norm of random matrices, approximation of covariance matrices, compressed sensing, restricted isometry property, log-concave random vectors, concentration inequalities, deviation inequalities, heavy tails, spectrum, singular values, order statistics 
@article{JEMS_2017_19_5_a4,
     author = {Olivier Gu\'edon and Alexander E. Litvak and Alain Pajor and Nicole Tomczak-Jaegermann},
     title = {On the interval of fluctuation of the singular values of random matrices},
     journal = {Journal of the European Mathematical Society},
     pages = {1469--1505},
     year = {2017},
     volume = {19},
     number = {5},
     doi = {10.4171/jems/697},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/697/}
}
TY  - JOUR
AU  - Olivier Guédon
AU  - Alexander E. Litvak
AU  - Alain Pajor
AU  - Nicole Tomczak-Jaegermann
TI  - On the interval of fluctuation of the singular values of random matrices
JO  - Journal of the European Mathematical Society
PY  - 2017
SP  - 1469
EP  - 1505
VL  - 19
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4171/jems/697/
DO  - 10.4171/jems/697
ID  - JEMS_2017_19_5_a4
ER  - 
%0 Journal Article
%A Olivier Guédon
%A Alexander E. Litvak
%A Alain Pajor
%A Nicole Tomczak-Jaegermann
%T On the interval of fluctuation of the singular values of random matrices
%J Journal of the European Mathematical Society
%D 2017
%P 1469-1505
%V 19
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4171/jems/697/
%R 10.4171/jems/697
%F JEMS_2017_19_5_a4
Olivier Guédon; Alexander E. Litvak; Alain Pajor; Nicole Tomczak-Jaegermann. On the interval of fluctuation of the singular values of random matrices. Journal of the European Mathematical Society, Tome 19 (2017) no. 5, pp. 1469-1505. doi: 10.4171/jems/697

Cité par Sources :