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 $$\mu $$ μ on $${\mathbb {R}}^n$$ R n and every $$\varepsilon > 0$$ ε > 0 , there exists a decomposition $$\mu = \int \mu _\theta d m(\theta )$$ μ = ∫ μ θ d m ( θ ) such that $$H(\mu ) - {\mathbb {E}}_{\theta \sim m} H(\mu _\theta ) \le \mathrm {Tr}(\mathrm {Cov}(\mu )) \varepsilon $$ H ( μ ) - E θ ∼ m H ( μ θ ) ≤ Tr ( Cov ( μ ) ) ε and $${\mathbb {E}}_{\theta \sim m} \mathrm {Cov}(\mu _\theta ) \preceq \mathrm {Id}/\varepsilon $$ E θ ∼ m Cov ( μ θ ) ⪯ Id / ε . 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 $$\{\pm \, 1 \}^n$$ { ± 1 } n or on $$[-\,1,1]^n$$ [ - 1 , 1 ] n , we show that the deficit between the mean-field approximation and the free energy is at most $$C \frac{1+p}{p} \left( n\Vert J\Vert _{S_p} \right) ^{\frac{p}{1+p}} $$ C 1 + p p n ‖ J ‖ S p p 1 + p for all $$p>0$$ p > 0 , where $$\Vert J\Vert _{S_p}$$ ‖ J ‖ S p denotes the Schatten- p norm of the interaction matrix. For the case $$p=2$$ 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.

中文翻译：

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通过熵有效的度量分解与平均场近似的应用来驯服相关性

统计物理学中各种模型的分析依赖于度量分解为类产品成分的混合物的存在，其中目标是实现分解为熵接近原始度量的度量，但之间的相关性很小坐标。我们证明了一个相关的一般结果：对于 $${\mathbb {R}}^n$$ R n 上的每个测度 $$\mu $$ μ 和每个 $$\varepsilon > 0$$ ε > 0 ，存在一个分解 $$\mu = \int \mu _\theta dm(\theta )$$ μ = ∫ μ θ dm ( θ ) 使得 $$H(\mu ) - {\mathbb {E}}_{\theta \sim m} H(\mu _\theta ) \le \mathrm {Tr}(\mathrm {Cov}(\mu )) \varepsilon $$ H ( μ ) - E θ ∼ m H ( μ θ ) ≤ Tr ( Cov ( μ ) ) ε 和 $${\mathbb {E}}_{\theta \sim m} \mathrm {Cov}(\mu _\theta ) \preceq \mathrm {Id}/\varepsilon $$ E θ ∼ m Cov ( μ θ ) ⪯ Id / ε 。作为应用程序，我们推导出 Ising 和 Potts 模型的平均场近似的一般界限，在连续和离散设置中，它在某种意义上都是无维数的。特别是，对于 $$\{\pm \, 1 \}^n$$ { ± 1 } n 或 $$[-\,1,1]^n$$ [ - 1 , 1 ] 上的 Ising 模型n ，我们表明平均场近似和自由能之间的赤字至多是 $$C \frac{1+p}{p} \left( n\Vert J\Vert _{S_p} \right) ^ {\frac{p}{1+p}} $$ C 1 + ppn ‖ J ‖ S pp 1 + p 对于所有 $$p>0$$ p > 0 ，其中 $$\Vert J\Vert _{S_p }$$ ‖ J ‖ S p 表示交互矩阵的 Schatten-p 范数。对于 $$p=2$$ p = 2 的情况，这恢复了 Jain 等人的结果。（平均场近似、凸层次结构和相关舍入的最优性：统一视角。arXiv:1808.07226 , 2018），