The statistical restricted isometry property and the Wigner semicircle distribution of incoherent dictionaries

TL;DR

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

Contribution

It develops a statistical RIP for incoherent dictionaries and links eigenvalue fluctuations to the Wigner semicircle distribution, with applications in 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 distribution.

Application to harmonic analysis dictionaries improves understanding of RIP in that setting.

Abstract

In this article we present 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 show 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.