Message Passing Algorithms for Compressed Sensing: I. Motivation and Construction

TL;DR

This paper introduces the derivation and extension of AMP algorithms for sparse signal reconstruction, connecting message passing techniques with belief propagation and statistical mechanics insights.

Contribution

It presents the first detailed derivation of AMP from belief propagation and explores its extensions and theoretical connections.

Findings

AMP algorithms effectively reconstruct sparse signals from limited measurements

Connections established between AMP, belief propagation, and statistical mechanics

Extensions improve the robustness and applicability of AMP methods

Abstract

In a recent paper, the authors proposed a new class of low-complexity iterative thresholding algorithms for reconstructing sparse signals from a small set of linear measurements \cite{DMM}. The new algorithms are broadly referred to as AMP, for approximate message passing. This is the first of two conference papers describing the derivation of these algorithms, connection with the related literature, extensions of the original framework, and new empirical evidence. In particular, the present paper outlines the derivation of AMP from standard sum-product belief propagation, and its extension in several directions. We also discuss relations with formal calculations based on statistical mechanics methods.