RSB Decoupling Property of MAP Estimators

TL;DR

This paper investigates the decoupling property of MAP estimators in large systems, showing that the joint distribution of input-output pairs converges to a scalar channel model under replica symmetry assumptions.

Contribution

It extends the decoupling principle to MAP estimators with a detailed analysis under replica symmetry breaking assumptions.

Findings

The joint distribution converges to a deterministic scalar channel model.

Under RS, interference terms vanish, recovering the known decoupling principle.

The analysis applies to various matrix ensembles and noise conditions.

Abstract

The large-system decoupling property of a MAP estimator is studied when it estimates the i.i.d. vector $\boldsymbol{x}$ from the observation $\boldsymbol{y}=\mathbf{A}\boldsymbol{x}+\boldsymbol{z}$ with $\mathbf{A}$ being chosen from a wide range of matrix ensembles, and the noise vector $\boldsymbol{z}$ being i.i.d. and Gaussian. Using the replica method, we show that the marginal joint distribution of any two corresponding input and output symbols converges to a deterministic distribution which describes the input-output distribution of a single user system followed by a MAP estimator. Under the $b$RSB assumption, the single user system is a scalar channel with additive noise where the noise term is given by the sum of an independent Gaussian random variable and $b$ correlated interference terms. As the $b$RSB assumption reduces to RS, the interference terms vanish which results in the formerly studied RS decoupling principle.