S-AMP: Approximate Message Passing for General Matrix Ensembles

TL;DR

This paper introduces S-AMP, an iterative estimation algorithm extending AMP to general matrix ensembles using free probability, with fixed points corresponding to stationary points of Gibbs free energy.

Contribution

It presents a novel S-AMP algorithm that generalizes AMP for arbitrary matrix spectra using the S-transform, ensuring optimality from its design.

Findings

S-AMP converges to stationary points of Gibbs free energy.

The algorithm is applicable to general matrix ensembles.

Optimality is inherent in the design of S-AMP.

Abstract

In this work we propose a novel iterative estimation algorithm for linear observation systems called S-AMP whose fixed points are the stationary points of the exact Gibbs free energy under a set of (first- and second-) moment consistency constraints in the large system limit. S-AMP extends the approximate message-passing (AMP) algorithm to general matrix ensembles. The generalization is based on the S-transform (in free probability) of the spectrum of the measurement matrix. Furthermore, we show that the optimality of S-AMP follows directly from its design rather than from solving a separate optimization problem as done for AMP.