Incoherent dictionaries and the statistical restricted isometry property

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, 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.

Key words: Incoherent dictionaries, statistical version of Candes - Tao RIP, Semi-Circle law, deterministic constructions, Heisenberg-Weil representation

References in corpus (3)

• Deterministic Designs with Deterministic Guarantees: Toeplitz Compressed Sensing Matrices, Sequence Designs and System Identification
• The finite harmonic oscillator and its associated sequences
• Bounds on certain Higher-Dimensional Exponential Sums via the Self-Reducibility of the Weil Representation

Cited by in corpus (5)

• Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property
• The statistical restricted isometry property and the Wigner semicircle distribution of incoherent dictionaries
• Deterministic Bounds for Restricted Isometry of Compressed Sensing Matrices
• Efficient Representations of Signals in Nonlinear Signal Processing with Applications to Inverse Problems
• Analog Coding Frame-work