Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property

TL;DR

This paper introduces a deterministic construction of sensing matrices satisfying a statistical isometry property, enabling efficient recovery of sparse signals with high probability, and broadening the scope of practical compressed sensing applications.

Contribution

It provides simple deterministic criteria for sensing matrices to act as near isometries on most sparse signals, with performance guarantees based on probabilistic signal models rather than matrix randomness.

Findings

Matrices formed from discrete chirps, Delsarte-Goethals codes, and extended BCH codes satisfy the criteria.

Recovery methods have expected performance sub-linear in signal dimension n.

Most sparse signals have unique representations in the measurement domain.

Abstract

Compressed Sensing aims to capture attributes of $k$-sparse signals using very few measurements. In the standard Compressed Sensing paradigm, the $\m\times \n$ measurement matrix $\A$ is required to act as a near isometry on the set of all $k$-sparse signals (Restricted Isometry Property or RIP). Although it is known that certain probabilistic processes generate $\m \times \n$ matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix $\A$ has this property, crucial for the feasibility of the standard recovery algorithms. In contrast this paper provides simple criteria that guarantee that a deterministic sensing matrix satisfying these criteria acts as a near isometry on an overwhelming majority of $k$-sparse signals; in particular, most such signals have a unique representation in the measurement domain. Probability still plays a critical role, but it enters the signal model rather than the construction of the sensing matrix. We require the columns of the sensing matrix to form a group under pointwise multiplication. The construction allows recovery methods for which the expected performance is sub-linear in $\n$, and only quadratic in $\m$; the focus on expected performance is more typical of mainstream signal processing than the worst-case analysis that prevails in standard Compressed Sensing. Our framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, Delsarte-Goethals codes, and extended BCH codes.