Bayes-Optimal Convolutional AMP

TL;DR

This paper introduces Bayes-optimal convolutional AMP (CAMP), a novel algorithm for compressed sensing that improves convergence and achieves Bayes-optimal performance for orthogonally invariant matrices by replacing the Onsager correction with a convolution of past messages.

Contribution

The paper develops a convolutional correction in AMP, derives an SE-based optimization for denoisers, and proves Bayes-optimality of CAMP under certain conditions for all orthogonally invariant matrices.

Findings

CAMP matches high-complexity AMP performance with simpler matched filter.

The SE equation guides optimal denoiser sequences for CAMP.

CAMP achieves Bayes-optimality under convergence and uniqueness conditions.

Abstract

This paper proposes Bayes-optimal convolutional approximate message-passing (CAMP) for signal recovery in compressed sensing. CAMP uses the same low-complexity matched filter (MF) for interference suppression as approximate message-passing (AMP). To improve the convergence property of AMP for ill-conditioned sensing matrices, the so-called Onsager correction term in AMP is replaced by a convolution of all preceding messages. The tap coefficients in the convolution are determined so as to realize asymptotic Gaussianity of estimation errors via state evolution (SE) under the assumption of orthogonally invariant sensing matrices. An SE equation is derived to optimize the sequence of denoisers in CAMP. The optimized CAMP is proved to be Bayes-optimal for all orthogonally invariant sensing matrices if the SE equation converges to a fixed-point and if the fixed-point is unique. For sensing matrices with low-to-moderate condition numbers, CAMP can achieve the same performance as high-complexity orthogonal/vector AMP that requires the linear minimum mean-square error (LMMSE) filter instead of the MF.