Taming correlations through entropy-efficient measure decompositions with applications to mean-field approximation

• 成果类型: Article
• 署名作者: Eldan, Ronen
• 署名单位: Weizmann Institute of Science
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00924-2
• 发表日期: 2020
• 页码: 737-755

摘要

The analysis of various models in statistical physics relies on the existence of decompositions of measures into mixtures of product-like components, where the goal is to attain a decomposition into measures whose entropy is close to that of the original measure, yet with small correlations between coordinates. We prove a related general result: For every measure mu on Rn and every epsilon>0, there exists a decomposition mu=integral mu theta dm(theta) such that H(mu)-E theta similar to mH(mu theta)<= Tr(Cov(mu))epsilon and E theta similar to mCov(mu theta)?Id/epsilon. As an application, we derive a general bound for the mean-field approximation of Ising and Potts models, which is in a sense dimension free, in both continuous and discrete settings. In particular, for an Ising model on {+/- 1}n or on [-1,1]n, we show that the deficit between the mean-field approximation and the free energy is at most C1+ppnJSpp1+p for all p>0, where JSp denotes the Schatten-p norm of the interaction matrix. For the case p=2, this recovers the result of Jain et al. (Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective. arXiv:1808.07226, 2018), but for an optimal choice of p it often allows to get almost dimension-free bounds.