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2015

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

Guédon, Olivier, Litvak, Alexander E., Pajor, Alain et al.

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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 $n/N$.

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

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