An Efficient Approach Toward the Asymptotic Analysis of Node-Based Recovery Algorithms in Compressed Sensing

TL;DR

This paper introduces a framework for analyzing the asymptotic performance of node-based recovery algorithms in compressed sensing, revealing success thresholds related to graph properties and demonstrating good agreement with finite simulations.

Contribution

It provides a novel asymptotic analysis framework for node-based algorithms in compressed sensing, identifying success thresholds and validating results with simulations.

Findings

Existence of a success threshold on the density ratio k/n

Asymptotic analysis aligns well with finite-length simulations

Recovery algorithms' success depends on graph and algorithm properties

Abstract

In this paper, we propose a general framework for the asymptotic analysis of node-based verification-based algorithms. In our analysis we tend the signal length $n$ to infinity. We also let the number of non-zero elements of the signal $k$ scale linearly with $n$. Using the proposed framework, we study the asymptotic behavior of the recovery algorithms over random sparse matrices (graphs) in the context of compressive sensing. Our analysis shows that there exists a success threshold on the density ratio $k/n$, before which the recovery algorithms are successful, and beyond which they fail. This threshold is a function of both the graph and the recovery algorithm. We also demonstrate that there is a good agreement between the asymptotic behavior of recovery algorithms and finite length simulations for moderately large values of $n$.