The Mutual Information in Random Linear Estimation

Understand

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 mutual information.