Restricted isometry property for random matrices with heavy-tailed columns

Abstract

Let A be a matrix whose columns X1,…,XN are independent random vectors in Rn. Assume that p-th moments of 〈Xi,a〉, a∈Sn−1, i⩽N, are uniformly bounded. For p>4, we prove that with high probability A has the Restricted Isometry Property (RIP) provided that Euclidean norms |Xi| are concentrated around n and that the covariance matrix is well approximated by the empirical covariance matrix provided that maxi|Xi|⩽C(nN)1/4. We also provide estimates for RIP when Eϕ(|〈Xi,a〉|)⩽1 for ϕ(t)=(1/2)exp(tα), with α∈(0,2].