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 $\beta >0$, the BA partition function $Z(x)=\int q^{\ast }(y)e^{-\beta |x-y{|}^2}dy$ satisfies $$Z(x)={\left(\frac{\pi }{\beta }\right)}^{d/2}p(x).$$ This identity, obtained from the BA fixed-point equation, implies that the BA effective score $g_{\beta }=-\nabla \log Z$ coincides exactly with the classical Fisher score $s=-\nabla \log p$ for all temperatures. Moreover, if $v=-\nabla \log q^{\ast }$ 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 $$J(p)=\mathcal{R}(v):=⟨v,\mathcal{G}v{⟩}_{L^2(q^{\ast })},$$ 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.

This is a noval explanation to information inequalities according to my paper "Relaxation kernel, spectral dissipation, and global convergence of {B}lahut--{A}rimoto dynamics" at arXiv:2604.25106