Matrix Fisher inequalities for non-invertible linear systems

The Fisher informatin matrix IX of a random vector X appears as useful information theoretic tool to describe the proppagation of information through systems. For instance, it is directly involved in the derivation of the Entropy Power Inequality (EPI), that describes the evolution of the entropy of random variables (vectors) submitted to linear transformations. The first results about information transformation were given in the 60’s by Blachman [1] and Stam [2]. Later, Papathanasiou [3] derived an important serie of Fisher Information Inequalities (FII) with applications to characterization of normality. In a fascinating contribution, Zamir [4] extended the FII to the case of noninvertible linear systems. He also pointed that such inequalities may have interesting applications in areas such as (blind) deconvolution and sources separation. However, the proofs given in his paper, completed in the technical report [5], involve complicated derivations, especially for the characterization of the cases of equality.