Replica Analysis for Generalized Linear Regression with IID Row Prior

TL;DR

This paper analyzes the MMSE for generalized linear models with IID row priors using the replica method, deriving exact solutions and proposing an efficient message passing algorithm validated by state evolution.

Contribution

It introduces a replica symmetric analysis for MMSE in models with IID row priors and develops a message passing algorithm with proven optimality in the asymptotic limit.

Findings

Exact MMSE derived via replica method

Proposed algorithm's MSE matches theoretical predictions

Algorithm attains optimal MSE if fixed point is unique

Abstract

Different from a typical independent identically distributed (IID) element assumption, this paper studies the estimation of IID row random matrix for the generalized linear model constructed by a linear mixing space and a row-wise mapping channel. The objective inference problem arises in many engineering fields, such as wireless communications, compressed sensing, and phase retrieval. We apply the replica method from statistical mechanics to analyze the exact minimum mean square error (MMSE) under the Bayes-optimal setting, in which the explicit replica symmetric solution of the exact MMSE estimator is obtained. Meanwhile, the input-output mutual information relation between the objective model and the equivalent single-vector system is established. To estimate the signal, we also propose a computationally efficient message passing based algorithm on expectation propagation (EP) perspective and analyze its dynamics. We verify that the asymptotic MSE of proposed algorithm predicted by its state evolution (SE) matches perfectly the exact MMSE estimator predicted by the replica method. That indicates, the optimal MSE error can be attained by the proposed algorithm if it has a unique fixed point.