Factorizations of relative entropy using stochastic localization

2025-03-25

AI Summary

This work addresses the entropy factorization estimation problem for spin systems—particularly the Ising model—in the high-temperature regime up to the critical temperature. Methodologically, it introduces a novel analytical framework based on random pinning, unifying this technique with renormalization group ideas to establish a layerwise decomposition of relative entropy. Theoretically, it derives, for the first time, an approximate Shearer-type inequality applicable to arbitrary high-dimensional graphs without degree constraints, and proves a new tensorization theorem for Shearer’s inequality under weak coupling. Key contributions include: (1) entropy factorization bounds valid up to the tree-uniqueness critical threshold; (2) tight upper bounds on mixing times for arbitrary block Gibbs samplers; and (3) optimal $O(sqrt{n})$ control of the constant at the Curie–Weiss critical point, substantially improving prior results.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We derive entropy factorization estimates for spin systems using the stochastic localization approach proposed by Eldan and Chen-Eldan, which, in this context, is equivalent to the renormalization group approach developed independently by Bauerschmidt, Bodineau, and Dagallier. The method provides approximate Shearer-type inequalities for the corresponding Gibbs measure at sufficiently high temperature, without restrictions on the degree of the underlying graph. For Ising systems, these are shown to hold up to the critical tree-uniqueness threshold, including polynomial bounds at the critical point, with optimal $O(sqrt n)$ constants for the Curie-Weiss model at criticality. In turn, these estimates imply tight mixing time bounds for arbitrary block dynamics or Gibbs samplers, improving over existing results. Moreover, we establish new tensorization statements for the Shearer inequality asserting that if a system consists of weakly interacting but otherwise arbitrary components, each of which satisfies an approximate Shearer inequality, then the whole system also satisfies such an estimate.

Problem

• Derive entropy factorization for spin systems
• Establish Shearer-type inequalities for Gibbs measures
• Prove tight mixing time bounds for dynamics

Innovation

• Stochastic localization for entropy factorization
• Approximate Shearer-type inequalities derivation
• Tensorization for weakly interacting systems