On the Convergence of Orthogonal/Vector AMP: Long-Memory Message-Passing Strategy

TL;DR

This paper proves the convergence of Bayes-optimal orthogonal/vector AMP algorithms in large systems using a novel long-memory message-passing approach, supported by theoretical analysis and numerical simulations.

Contribution

It introduces a new long-memory message-passing framework that guarantees convergence of orthogonal/vector AMP in the large system limit.

Findings

Convergence of Bayes-optimal orthogonal/vector AMP is established.

A new statistical interpretation of LM damping is proposed.

Numerical results verify theoretical state evolution predictions.

Abstract

Orthogonal/vector approximate message-passing (AMP) is a powerful message-passing (MP) algorithm for signal reconstruction in compressed sensing. This paper proves the convergence of Bayes-optimal orthogonal/vector AMP in the large system limit. The proof strategy is based on a novel long-memory (LM) MP approach: A first step is a construction of LM-MP that is guaranteed to converge systematically. A second step is a large-system analysis of LM-MP via an existing framework of state evolution. A third step is to prove the convergence of state evolution recursions for Bayes-optimal LM-MP via a new statistical interpretation of existing LM damping. The last is an exact reduction of the state evolution recursions for Bayes-optimal LM-MP to those for Bayes-optimal orthogonal/vector AMP. The convergence of the state evolution recursions for Bayes-optimal LM-MP implies that for Bayes-optimal orthogonal/vector AMP. Numerical simulations are presented to show the verification of state evolution results for damped orthogonal/vector AMP and a negative aspect of LM-MP in finite-sized systems.