Density Evolution Analysis of Node-Based Verification-Based Algorithms in Compressed Sensing

TL;DR

This paper introduces a simplified and accurate asymptotic analysis method for node-based verification algorithms in compressed sensing, predicting their performance as the signal dimension grows large.

Contribution

It develops a density evolution-based analysis for NB-VB algorithms, adapting message-passing techniques without extrinsic information, improving simplicity and accuracy over existing methods.

Findings

Analysis accurately predicts unverified signal fraction at each iteration

Performance closely matches simulations for large signal dimensions

Method is simpler and more precise than differential equations approach

Abstract

In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressive sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension $n$). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration $\ell$ in the asymptotic regime where $n \rightarrow \infty$. The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of non-trivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of $\ell$. Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of $n$. Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is much simpler and more accurate.