Statistical RIP and Semi-Circle Distribution of Incoherent Dictionaries

In this paper we formulate and prove a statistical version of the Candès-Tao restricted isometry property (SRIP for short) which holds in general for any incoherent dictionary which is a disjoint union of orthonormal bases. In addition, we prove that, under appropriate normalization, the eigenvalues of the associated Gram matrix fluctuate around λ = 1 according to the Wigner semicircle distribution. The result is then applied to various dictionaries that arise naturally in the setting of finite harmonic analysis, giving, in particular, a better understanding on a remark of Applebaum-Howard-SearleCalderbank concerning RIP for the Heisenberg dictionary of chirp like functions.