On Log-Concave-Tailed Chaoses and the Restricted Isometry Property

Abstract: In this paper, we obtain a $p$-th moment bound for the suprema of a log-concave-tailed nonhomogeneous chaos process, which is optimal in some special cases. A crucial ingredient of the proof is a novel decoupling inequality, which may be of independent interest. With this $p$-th moment bound, we show two uniform Hanson-Wright type deviation inequalities for $\alpha$-subexponential entries ($1\le \alpha\le 2$), which recover some known results. As applications, we prove the restricted isometry property of partial random circulant matrices and time-frequency structured random matrices induced by standard $\alpha$-subexponential vectors ($1\le \alpha\le 2$), which extends the previously known results for the subgaussian case.