I-MMSE relations in random linear estimation and a sub-extensive interpolation method

Abstract: Consider random linear estimation with Gaussian measurement matrices and noise. One can compute infinitesimal variations of the mutual information under infinitesimal variations of the signal-to-noise ratio or of the measurement rate. We discuss how each variation is related to the minimum mean-square error and deduce that the two variations are directly connected through a very simple identity. The main technical ingredient is a new interpolation method called "sub-extensive interpolation method". We use it to provide a new proof of an I-MMSE relation recently found by Reeves and Pfister [1] when the measurement rate is varied. Our proof makes it clear that this relation is intimately related to another I-MMSE relation also recently proved in [2]. One can directly verify that the identity relating the two types of variation of mutual information is indeed consistent with the one letter replica symmetric formula for the mutual information, first derived by Tanaka [3] for binary signals, and recently proved in more generality in 1,2,4,5. However our proof is independent of any knowledge of Tanaka's formula.