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.

Voir la notice de l'article provenant de la source EMS Press

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 
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     title = {On the interval of fluctuation of the singular values of random matrices},
     journal = {Journal of the European Mathematical Society},
     pages = {1469--1505},
     publisher = {mathdoc},
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     number = {5},
     year = {2017},
     doi = {10.4171/jems/697},
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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. http://geodesic.mathdoc.fr/articles/10.4171/jems/697/

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