Suprema of Chaos Processes and the Restricted Isometry Property

Abstract

We present a new bound for suprema of a special type of chaos processes indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and time-frequency structured random matrices. In both cases the required condition on the number $m$ of rows in terms of the sparsity $s$ and the vector length $n$ is $m \gtrsim s \log^2 s \log^2 n$.