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

2026-06-04

AI Summary

This work uncovers the intrinsic geometric structure linking Fisher information and the entropy power inequality within Blahut–Arimoto (BA) rate–distortion optimization. By analyzing the translational symmetry inherent in the BA framework, the study reveals for the first time that Fisher information acts as a temperature-invariant spectral quantity—interpreted as a “spectral charge” associated with translational modes. Through variational analysis combined with tools from information geometry, the authors rigorously establish that Fisher information equals the Rayleigh quotient corresponding to these translational modes. This identity naturally yields the Fisher information inequality and, as its classical corollary, the entropy power inequality. The results provide a novel geometric perspective and a unified conceptual framework for understanding fundamental information-theoretic inequalities.

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.

Problem

• Translation Symmetry
• Fisher Information
• Entropy Power Inequality
• Blahut–Arimoto Geometry
• Rate-Distortion

Innovation

• Translation Symmetry
• Fisher Information
• Blahut–Arimoto Algorithm
• Entropy Power Inequality
• Rayleigh Quotient