Analysis of the Statistical Restricted Isometry Property for Deterministic Sensing Matrices Using Stein’s Method

Statistical restricted isometry property (STRIP) was recently formulated by Calderbank et al. to analyze the performance of deterministic sampling matrices for compressed sensing. In this paper, we study the STRIP by taking advantage of concentration inequalities using Stein’s method. In particular, we derive the STRIP performance bound in terms of the mutual coherence of the sampling matrix and the sparsity level of the input signal. Based on such connections, we show that a large class of deterministic matrices can satisfy the STRIP with high probability provided that they can nearly meet the Welch bound. Such matrices include many classical spreading codes in codedivision multiple access (CDMA) such as the Kasami code, the Gold code, the Frank-Zadoff-Chu code, and the more recent partial FFT matrices based on difference sets etc. Simulation results show that these deterministic sensing matrices can offer reconstruction performance similar to that of random operators.