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

Abstract

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 ${\mathbb {R}}^n$ and every $\varepsilon > 0$, there exists a decomposition $\mu = \int \mu _\theta d m(\theta )$ such that $H(\mu ) - {\mathbb {E}}_{\theta \sim m} H(\mu _\theta ) \le \mathrm {Tr}(\mathrm {Cov}(\mu )) \varepsilon$ and ${\mathbb {E}}_{\theta \sim m} \mathrm {Cov}(\mu _\theta ) \preceq \mathrm {Id}/\varepsilon$. 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 $\{\pm \, 1 \}^n$ or on $[-\,1,1]^n$, we show that the deficit between the mean-field approximation and the free energy is at most $C \frac{1+p}{p} \left( n\Vert J\Vert _{S_p} \right) ^{\frac{p}{1+p}}$ for all $p>0$, where $\Vert J\Vert _{S_p}$ 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.