State Evolution for General Approximate Message Passing Algorithms, with Applications to Spatial Coupling

TL;DR

This paper analyzes the high-dimensional behavior of a broad class of approximate message passing algorithms using state evolution, with applications to spatially coupled sensing matrices in compressed sensing.

Contribution

It introduces a generalized state evolution framework for AMP algorithms applicable to non-i.i.d. Gaussian matrices, simplifying and extending previous proofs.

Findings

Characterizes AMP behavior with state evolution for non-i.i.d. Gaussian matrices

Applies analysis to spatially coupled sensing matrices in compressed sensing

Simplifies and generalizes existing proof techniques

Abstract

We consider a class of approximated message passing (AMP) algorithms and characterize their high-dimensional behavior in terms of a suitable state evolution recursion. Our proof applies to Gaussian matrices with independent but not necessarily identically distributed entries. It covers --in particular-- the analysis of generalized AMP, introduced by Rangan, and of AMP reconstruction in compressed sensing with spatially coupled sensing matrices. The proof technique builds on the one of [BM11], while simplifying and generalizing several steps.