Statistical RIP and Semi-Circle Distribution of Incoherent Dictionaries

Details Statistical RIP and Semi-Circle Distribution of Incoherent Dictionaries Statistical RIP and Semi-Circle Distribution of Incoherent Dictionaries

2009-03-20

arxiv.org/abs/0903.3627v1

Computer Science

Information Theory

This version Includes all the proofs. Submitted for publication (2009)

Scientific paper

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.

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