State-Evolution-based Score Matching for Generalized Approximate Message Passing

2026-06-28

AI Summary

This work addresses the limited applicability of Bayes-GAMP in complex-valued settings and under arbitrary nonlinear observation models, where the required posterior denoisers typically lack closed-form expressions. By characterizing the message-passing dynamics of GAMP through state evolution, the authors introduce a score-matching framework to train a neural network that replaces analytically intractable denoisers. This approach enables approximate Bayesian-optimal inference using only forward evaluations of the observation mapping, without requiring explicit knowledge of its functional form. Notably, it is the first method to support nearly arbitrary complex-valued nonlinear observation models, eliminating dependence on closed-form denoisers or explicit model specifications. Under ideal training conditions, the proposed algorithm asymptotically approaches the performance of the true Bayes-GAMP, substantially broadening the scope of GAMP-based inference.

Abstract

Generalized approximate message passing (GAMP) equipped with minimum mean-square error (MMSE) denoisers, commonly referred to as Bayes-GAMP, is a powerful framework for solving inverse problems described by generalized linear models (GLMs) with arbitrary component-wise nonlinearities in the observation process. However, despite its theoretical tractability and rigorously established asymptotic optimality, the range of practical observation models for which Bayes-GAMP admits a closed-form implementation remains severely limited, particularly in complex-valued settings. This limitation largely stems from the restrictive requirement that the corresponding output denoiser, given by a conditional expectation, admit a closed-form expression. To overcome this limitation, we propose a principled approach that enables the implementation of Bayes-GAMP for complex-valued models with \emph{virtually arbitrary} nonlinear observation mappings. Specifically, within a score-matching framework, we train a neural network to emulate the output denoiser using training data generated from a characterization of the message dynamics based on state evolution (SE). Notably, the proposed approach requires neither explicit evaluation of the denoiser nor knowledge of an explicit functional form of the nonlinear mapping; it requires only access to forward evaluations of the mapping during offline training. We show that, under ideal training conditions, GAMP with the trained network replacing the analytically intractable denoiser asymptotically matches the performance of Bayes-GAMP with the exact denoiser.

Problem

• Generalized Approximate Message Passing
• Bayes-GAMP
• complex-valued models
• nonlinear observation mappings
• denoiser

Innovation

• State Evolution
• Score Matching
• Generalized Approximate Message Passing
• Neural Denoiser
• Complex-valued Inverse Problems