Sufficient Statistic Memory Approximate Message Passing

TL;DR

This paper introduces SS-MAMP, a memory-augmented AMP algorithm that guarantees convergence by using sufficient statistics, ensuring accurate state evolution description and preserving orthogonality.

Contribution

It proposes SS-MAMP, a novel AMP variant with convergence guarantees via sufficient statistics, extending the applicability of AMP algorithms.

Findings

Covariance matrices of SS-MAMP are L-banded and convergent.

Damping can construct SS-MAMP from any MAMP, ensuring convergence.

SS-MAMP preserves orthogonality and accurate state evolution.

Abstract

Approximate message passing (AMP) type algorithms have been widely used in the signal reconstruction of certain large random linear systems. A key feature of the AMP-type algorithms is that their dynamics can be correctly described by state evolution. However, state evolution does not necessarily guarantee the convergence of iterative algorithms. To solve the convergence problem of AMP-type algorithms in principle, this paper proposes a memory AMP (MAMP) under a sufficient statistic condition, named sufficient statistic MAMP (SS-MAMP). We show that the covariance matrices of SS-MAMP are L-banded and convergent. Given an arbitrary MAMP, we can construct the SS-MAMP by damping, which not only ensures the convergence, but also preserves the orthogonality, i.e., its dynamics can be correctly described by state evolution.