Generalized Memory Approximate Message Passing

TL;DR

This paper introduces a universal GMAMP framework that unifies existing AMP algorithms and achieves Bayes-optimal signal reconstruction for unitarily-invariant matrices with low complexity.

Contribution

It proposes a new GMAMP framework encompassing existing algorithms and develops a Bayes-optimal variant with low complexity for generalized linear models.

Findings

GMAMP unifies existing AMP algorithms.

BO-GMAMP achieves Bayes-optimal MSE.

Framework guarantees asymptotic Gaussianity and state evolution.

Abstract

Generalized approximate message passing (GAMP) is a promising technique for unknown signal reconstruction of generalized linear models (GLM). However, it requires that the transformation matrix has independent and identically distributed (IID) entries. In this context, generalized vector AMP (GVAMP) is proposed for general unitarily-invariant transformation matrices but it has a high-complexity matrix inverse. To this end, we propose a universal generalized memory AMP (GMAMP) framework including the existing orthogonal AMP/VAMP, GVAMP, and memory AMP (MAMP) as special instances. Due to the characteristics that local processors are all memory, GMAMP requires stricter orthogonality to guarantee the asymptotic IID Gaussianity and state evolution. To satisfy such orthogonality, local orthogonal memory estimators are established. The GMAMP framework provides a principle toward building new advanced AMP-type algorithms. As an example, we construct a Bayes-optimal GMAMP (BO-GMAMP), which uses a low-complexity memory linear estimator to suppress the linear interference, and thus its complexity is comparable to GAMP. Furthermore, we prove that for unitarily-invariant transformation matrices, BO-GMAMP achieves the replica minimum (i.e., Bayes-optimal) MSE if it has a unique fixed point.