Translation Symmetry, Fisher Information, and the Entropy Power Inequality in Blahut--Arimoto Geometry

Abstract

We identify a previously unrecognised structure in the finite-temperature geometry of Blahut--Arimoto (BA) rate-distortion optimisation. The starting point is an exact partition identity. For every source density (p) and every inverse temperature $β>0$, the BA partition function $Z(x)=\int q^*(y)e^{-β|x-y|^2}dy$ satisfies $$ Z(x)=\left(\fracπβ\right)^{d/2}p(x). $$ This identity, obtained from the BA fixed-point equation, implies that the BA effective score $g_β=-\nabla\log Z$ coincides exactly with the classical Fisher score $s=-\nabla\log p$ for all temperatures. Moreover, if $v=-\nabla\log q^*$ denotes the translation mode generated by the quadratic-distortion symmetry, then its BA projection satisfies $\mathcal P v=-s$. These observations lead to the central identity $$ J(p)=\mathcal R(v):=\langle v,\mathcal G v\rangle_{L^2(q^*)}, $$ where $\mathcal G$ is the BA relaxation kernel. Thus Fisher information is exactly the Rayleigh quotient of the translation mode and is therefore a temperature-invariant spectral quantity in the BA framework. This yields a geometric interpretation of the Fisher information inequality: the inequality $$ J(X+Y)^{-1}\ge J(X)^{-1}+J(Y)^{-1} $$ becomes the parallel-combination law of a Rayleigh quotient under convolution. The entropy power inequality then follows through the standard heat-flow argument. The contribution is not a new proof of the entropy power inequality, but the identification of a hidden geometric structure: Fisher information as the spectral charge of the translation mode in BA rate-distortion geometry, with the entropy power inequality emerging as a consequence of this temperature-invariant fact.