Statistical RIP and Semi-Circle Distribution of Incoherent Dictionaries

TL;DR

This paper introduces a statistical version of the restricted isometry property for incoherent dictionaries and demonstrates that their Gram matrix eigenvalues follow the Wigner semicircle distribution, enhancing understanding in harmonic analysis.

Contribution

It formulates and proves a statistical RIP for incoherent dictionaries and shows eigenvalues follow the semicircle distribution, connecting random matrix theory with harmonic analysis.

Findings

Statistical RIP holds for incoherent dictionaries composed of orthonormal bases.

Eigenvalues of Gram matrices fluctuate around 1 according to the Wigner semicircle law.

Application to harmonic analysis dictionaries improves understanding of RIP in this context.

Abstract

In this paper we formulate and prove a statistical version of the Candes-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 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-Searle-Calderbank concerning RIP for the Heisenberg dictionary of chirp like functions.