  The large-system performance of MAP 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 (RS) 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 MAP 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 linear systems followed by MAP estimators. The structure
of the decoupled linear system is further studied under both the RS and the
Replica Symmetry Breaking (RSB) assumptions. For $b$ steps of RSB, the
decoupled system is found to be an additive system with a noise term given as
the sum of an independent Gaussian random variable with $b$ correlated
impairment terms. As an application of our study, we investigate large
compressive sensing systems by considering the $\ell_p$ minimization recovery
schemes. Our numerical investigations show that the replica symmetric ansatz
for $\ell_0$ norm recovery fails to give an accurate approximation of the mean
square error as the compression rate grows, and therefore, the RSB ans\"atze
are needed.
