Entropic independence via sparse localization

2026-04-12

AI Summary

Existing entropy independence criteria often rely on spectral independence or uniform bounds over all conditional marginals, which may fail in regular ensemble models exhibiting strong mixing properties. This work proposes a sparse localization framework that establishes quadratic entropy stability and entropy independence by verifying $\ell^2$-independence only for a sparse family of conditionals where at most $cn$ coordinates are fixed. The approach incurs a multiplicative loss of order $c^{-1}$ and circumvents the need for global conditional assumptions. It is the first to achieve entropy independence under such sparse conditions, significantly broadening applicability. As a concrete application, the framework rigorously proves approximate entropy preservation for the uniform distribution over independent sets of fixed size in bounded-degree graphs.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Entropic independence is a structural property of measures that underlies modern proofs of functional inequalities, notably (modified) log-Sobolev inequalities, via ``annealing'' or local-to-global schemes. Existing sufficient criteria for entropic independence typically require spectral independence and/or uniform bounds on marginals under \emph{all} pinnings, which can fail in natural canonical-ensemble models even when strong mixing properties are expected. We introduce \emph{sparse localization}: a restricted localization framework, in the spirit of Chen--Eldan, in which one assumes $\ell_2$-independence only for a sparse family of pinnings (those fixing at most $cn$ coordinates for any $c > 0$), yet still deduces quadratic entropic stability and entropic independence with an explicit multiplicative loss of order $c^{-1}$. As an application, we give a rigorous proof of approximate conservation of entropy for the uniform distribution on independent sets of a given size in bounded degree graphs.

Problem

• entropic independence
• spectral independence
• canonical-ensemble models
• marginals under pinnings
• strong mixing properties

Innovation

• sparse localization
• entropic independence
• ℓ²-independence
• log-Sobolev inequalities
• canonical ensemble