  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] (by independent methods).
However our proof is independent of any knowledge of Tanaka's formula.
