There has been definite progress recently in proving the variational single-letter formula
given by the heuristic replica method for various estimation problems. In particular, the
replica formula for the mutual information in the case of noisy linear estimation with random
iid matrices, a problem with applications ranging from compressed sensing to statistics, has
been proven rigorously. In this contribution we go beyond the restrictive iid matrix
assumption and discuss the formula proposed by Takeda, Uda, Kabashima and later by …

We consider the estimation of a signal from the knowledge of its noisy linear random
Gaussian projections, a problem relevant in compressed sensing, sparse superposition
codes or code division multiple access just to cite few. There has been a number of works
considering the mutual information for this problem using the heuristic replica method from
statistical physics. Here we put these considerations on a firm rigorous basis. First, we show,
using a Guerra-type interpolation, that the replica formula yields an upper bound to the exact …