Approximate Message Passing with Unitary Transformation

TL;DR

This paper introduces UT-AMP, a new variant of approximate message passing that uses a unitary transformation to ensure convergence for any matrix A, significantly improving robustness and speed over traditional AMP.

Contribution

The paper develops UT-AMP, a unitary transformation-based AMP variant that guarantees convergence for any matrix A, extending previous work and providing theoretical analysis.

Findings

UT-AMP always converges for Gaussian priors regardless of matrix A.

UT-AMP demonstrates much greater robustness than original AMP.

UT-AMP exhibits fast convergence in various scenarios.

Abstract

Approximate message passing (AMP) and its variants, developed based on loopy belief propagation, are attractive for estimating a vector x from a noisy version of z = Ax, which arises in many applications. For a large A with i. i. d. elements, AMP can be characterized by the state evolution and exhibits fast convergence. However, it has been shown that, AMP mayeasily diverge for a generic A. In this work, we develop a new variant of AMP based on a unitary transformation of the original model (hence the variant is called UT-AMP), where the unitary matrix is available for any matrix A, e.g., the conjugate transpose of the left singular matrix of A, or a normalized DFT (discrete Fourier transform) matrix for any circulant A. We prove that, in the case of Gaussian priors, UT-AMP always converges for any matrix A. It is observed that UT-AMP is much more robust than the original AMP for difficult A and exhibits fast convergence. A special form of UT-AMP with a circulant A was used in our previous work [13] for turbo equalization. This work extends it to a generic A, and provides a theoretical investigation on the convergence.