Complex Approximate Message Passing with Non-separable Denoising

2026-04-22

AI Summary

This work addresses the absence of a unified state evolution theory for approximate message passing (AMP) in the complex domain with non-separable denoisers. By constructing an augmented real-valued system and introducing a many-to-one canonical transformation, the authors embed complex non-separable denoising into a higher-dimensional real space, process it therein, and then map back—thereby establishing, for the first time, a rigorous AMP state evolution framework for complex-valued non-separable settings. This formulation naturally yields an Onsager correction term expressed via Wirtinger derivatives and extends seamlessly to matrix-valued problems, enabling joint recovery of multiple eigenvectors. Experiments demonstrate that the proposed method significantly outperforms both separable and real-valued AMP in tasks such as OTFS-based grant-free random access preamble detection, with the state evolution accurately predicting empirical algorithm performance.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Approximate Message Passing
• Complex-valued inference
• Non-separable denoising
• State evolution
• Compressed sensing

Innovation

• Complex AMP
• Non-separable Denoising
• State Evolution
• Wirtinger Derivatives
• Sparse Group LASSO