Information loss under coarse-graining: a geometric approach

TL;DR

This paper employs information geometry to analyze how information about relevant and irrelevant parameters evolves under coarse-graining, providing a geometric understanding of universality and the flow of models in statistical systems.

Contribution

It introduces a covariant geometric formalism to quantify information loss and preservation during renormalization group flow, linking information theory with statistical physics.

Findings

Relevant parameters retain their distinguishability under flow.

Irrelevant parameters become less distinguishable, contracting according to RG exponents.

The approach explains universality through information-theoretic flow of the model manifold.

Abstract

We use information geometry, in which the local distance between models measures their distinguishability from data, to quantify the flow of information under the renormalization group. We show that information about relevant parameters is preserved, with distances along relevant directions maintained under flow. By contrast, irrelevant parameters become less distinguishable under the flow, with distances along irrelevant directions contracting according to renormalization group exponents. We develop a covariant formalism to understand the contraction of the model manifold. We then apply our tools to understand the emergence of the diffusion equation and more general statistical systems described by a free energy. Our results give an information-theoretic justification of universality in terms of the flow of the model manifold under coarse-graining.