Limits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices

TL;DR

This paper introduces a probabilistic framework for analyzing the restricted isometry constant (RIC) of finite Gaussian matrices in compressed sensing, providing bounds on sparse recovery limits and approximations using Tracy-Widom distribution.

Contribution

It offers a new approach based on Wishart eigenvalue distributions to analyze RICs and recovery limits for finite Gaussian measurement matrices in compressed sensing.

Findings

Derived probability bounds for RIC satisfaction in finite Gaussian matrices.

Provided tight lower bounds on maximum sparsity for successful recovery.

Developed simple RIC approximations using Tracy-Widom distribution.

Abstract

One of the key issues in the acquisition of sparse data by means of compressed sensing (CS) is the design of the measurement matrix. Gaussian matrices have been proven to be information-theoretically optimal in terms of minimizing the required number of measurements for sparse recovery. In this paper we provide a new approach for the analysis of the restricted isometry constant (RIC) of finite dimensional Gaussian measurement matrices. The proposed method relies on the exact distributions of the extreme eigenvalues for Wishart matrices. First, we derive the probability that the restricted isometry property is satisfied for a given sufficient recovery condition on the RIC, and propose a probabilistic framework to study both the symmetric and asymmetric RICs. Then, we analyze the recovery of compressible signals in noise through the statistical characterization of stability and robustness. The presented framework determines limits on various sparse recovery algorithms for finite size problems. In particular, it provides a tight lower bound on the maximum sparsity order of the acquired data allowing signal recovery with a given target probability. Also, we derive simple approximations for the RICs based on the Tracy-Widom distribution.