  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.
