Complex Approximate Message Passing with Non-separable Denoising

TL;DR

This paper develops a unified state evolution theory for complex Approximate Message Passing (AMP) algorithms with non-separable denoisers, enabling improved high-dimensional complex-valued inference.

Contribution

It introduces a framework that extends AMP to complex, non-separable denoisers using an augmented real-valued system and Wirtinger derivatives, unifying previous approaches.

Findings

State evolution accurately predicts performance of complex AMP with non-separable denoisers.

Complex non-separable denoising outperforms separable and real-valued methods.

Framework extends to matrix-valued settings, enabling joint structural constraints exploitation.

Abstract

Approximate Message Passing (AMP) is a general framework for iterative algorithms, originally developed for compressed sensing and later extended to a wide range of high-dimensional inference problems. Although recent work has advanced matrix AMP, complex AMP, and AMP for non-separable functions independently, a unified state evolution theory for complex AMP with non-separable denoisers has been lacking. This article fills that gap by establishing state evolution in the setting of complex, non-separable denoising functions. The proposed approach constructs an augmented real-valued system that lifts the problem to a higher-dimensional space, then recovers the complex domain through a many-to-one canonical transformation. Under this construction, the Onsager correction naturally involves Wirtinger derivatives, and the resulting state evolution reduces to scalar complex recursions despite the non-separable structure of the denoisers. The framework extends to the matrix-valued setting, accommodating multiple feature vectors simultaneously. This generalization enables AMP to exploit joint structural constraints, such as simultaneous group and element sparsity, in complex-valued recovery problems. The complex sparse group least absolute shrinkage and selection operator (LASSO) serves as a key instantiation, motivated by preamble detection in Orthogonal Time-Frequency Space (OTFS)-based unsourced random access. Numerical experiments confirm that state evolution accurately predicts performance and show that complex non-separable denoising can produce significant gains over separable and real-valued alternatives.