Statistical Mechanics of MAP Estimation: General Replica Ansatz

The large-system performance of maximum-a-poste-rior estimation is studied considering a general distortion function when the observation vector is received through a linear system with additive white Gaussian noise. The analysis considers the system matrix to be chosen from the large class of rotationally invariant random matrices. We take a statistical mechanical approach by introducing a spin glass corresponding to the estimator, and employing the replica method for the large-system analysis. In contrast to earlier replica based studies, our analysis evaluates the general replica ansatz of the corresponding spin glass and determines the asymptotic distortion of the estimator for any structure of the replica correlation matrix. Consequently, the replica symmetric as well as the replica symmetry breaking ansatz with b steps of breaking is deduced from the given general replica ansatz. The generality of our distortion function lets us derive a more general form of the maximum-a-posterior decoupling principle. Based on the general replica ansatz, we show that for any structure of the replica correlation matrix, the vector-valued system decouples into a bank of equivalent decoupled scalar systems followed by maximum-a-posterior estimators. The structure of the decoupled system is further studied under both the replica symmetry and the replica symmetry breaking assumptions. For b steps of symmetry breaking, the decoupled system is found to be an additive system with a non-Gaussian noise term given as the sum of an independent Gaussian random variable with b non-Gaussian impairment terms which depend on the input symbol. The general decoupling property of the maximum-a-posterior estimator leads to the idea of a replica simulator which represents the replica ansatz through the state evolution of a transition system described by its corresponding decoupled system. As an application of our study, we investigate large compressive sensing systems by considering the \ell _{p} norm minimization recovery schemes. Our numerical investigations show that the replica symmetric ansatz for \ell _{0} norm recovery fa