Sufficient-Statistic Memory AMP

TL;DR

This paper introduces the SS-MAMP framework, ensuring convergence of AMP-type algorithms' state evolution under certain conditions, and demonstrates its Bayes-optimality and practical effectiveness through theoretical proofs and simulations.

Contribution

It proposes a sufficient-statistic memory AMP framework that guarantees convergence and preserves optimality, extending the applicability of AMP algorithms.

Findings

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

Constructed SS-BO-MAMP achieves Bayes-optimal MSE predicted by replica methods.

Simulations confirm the theoretical convergence and performance improvements.

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. While state evolution is a useful analytic tool, its convergence is not guaranteed. To solve the convergence problem of the state evolution of AMP-type algorithms in principle, this paper proposes a sufficient-statistic memory AMP (SS-MAMP) algorithm framework under the conditions of right-unitarily invariant sensing matrices, Lipschitz-continuous local processors and the sufficient-statistic constraint (i.e., the current message of each local processor is a sufficient statistic of the signal vector given the current and all preceding messages). We show that the covariance matrices of SS-MAMP are L-banded and convergent, which is an optimal framework (from the local MMSE/LMMSE perspective) for AMP-type algorithms given the Lipschitz-continuous local processors. Given an arbitrary MAMP, we can construct an SS-MAMP by damping, which not only ensures the convergence of the state evolution, but also preserves the orthogonality, i.e., its dynamics can be correctly described by state evolution. As a byproduct, we prove that the Bayes-optimal orthogonal/vector AMP (BO-OAMP/VAMP) is an SS-MAMP. As an example, we construct a sufficient-statistic Bayes-optimal MAMP (SS-BO-MAMP) whose state evolution converges to the minimum (i.e., Bayes-optimal) mean square error (MSE) predicted by replica methods when it has a unique fixed point. In addition, the MSE of SS-BO-MAMP is not worse than the original BO-MAMP. Finally, simulations are provided to support the theoretical results.