This is an announcement for the paper "On the interval of fluctuation of the singular values of random" by Olivier Guedon, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann.
Abstract: Let $A$ be a matrix whose columns $X_1,\dots, X_N$ are independent random vectors in $\mathbb{R}^n$. Assume that the tails of the 1-dimensional marginals decay as $\mathbb{P}(|\langle X_i, a\rangle|\geq t)\leq t^{-p}$ uniformly in $a\in S^{n-1}$ and $i\leq N$. Then for $p>4$ we prove that with high probability $A/{\sqrt{n}}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\sqrt{n}$. We also show that the covariance matrix is well approximated by the empirical covariance matrix 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({-t^{\alpha}})$ with $\alpha \in (0,2]$, extending the known case $\alpha\in[1, 2]$.
Archive classification: math.PR cs.IT math.FA math.IT
Mathematics Subject Classification: 60B20, 46B06, 15B52, 46B09, 60D05
Remarks: To appear in J. Eur. Math. Soc
Submitted from: olivier.guedon@univ-mlv.fr
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1509.02322
or