Replica Analysis for Generalized Linear Regression with IID Row Prior.

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.