Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations
Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations
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DOI：
10.1007/s00039-018-0461-z
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发表时间：
2016-12
影响因子：
2.2
通讯作者：
Ronen Eldan
中科院分区：
文献类型：
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作者：
Ronen Eldan
我们证明了离散立方体和高斯空间上的测度的结构定理，这为平均场行为提供了充分的条件。这些条件依赖于此类度量的新复杂性概念，即对数密度梯度的高斯宽度。在立方体 {−1, 1}n 上，我们表明，表现出低复杂性的测度可以写成测度的混合，这样：（i）对于每个测度，测度是一个小扰动，使得 log 是一个梯度很小的线性函数，（ii）对于大多数来说，在 Wasserstein 距离中接近于某个乘积测度。因此，我们的框架可用于研究除配分函数近似之外的低复杂性度量的行为，表明这些度量大致是其熵接近原始度量的乘积度量的混合物。特别是，作为我们定理的推论，我们推导了对数配分函数的朴素平均场近似的界限，该界限改进了 Chatterjee 和 Dembo 的非线性大偏差框架（Adv Math，319：313-347，2017。ISSN 0001-8708。 https://dx.doi.org/10.1016/j.aim.2017.08.003 ）以多种方式：（1）它不需要二阶导数的任何界限。 (2) 覆盖数被较弱的高斯宽度概念取代。 (3) 我们获得了关于维度的更强渐近性。另外两个推论是指数随机图和大阶 Ising 模型的分解定理。在高斯情况下，我们表明低复杂性的度量表现出几乎严格的逆对数索博列夫不等式。
We prove structure theorems for measures on the discrete cube and on Gaussian space, which provide sufficient conditions for mean-field behavior. These conditions rely on a new notion of complexity for such measures, namely the Gaussian-width of the gradient of the log-density. On the cube {−1, 1}n, we show that a measurewhich exhibits low complexity can be written as a mixtureof measuressuch that: (i) for each, the measureis a small perturbation ofsuch that logis a linear function whose gradient is small and, (ii)is close to some product measure, in Wasserstein distance, for most. Thus, our framework can be used to study the behavior of low-complexity measures beyond approximation of the partition function, showing that those measures are roughly mixtures of product measures whose entropy is close to that of the original measure. In particular, as a corollary of our theorems, we derive a bound for the naïve mean-field approximation of the log-partition function which improves the nonlinear large deviation framework of Chatterjee and Dembo (Adv Math, 319:313–347, 2017. ISSN 0001-8708. https://dx.doi.org/10.1016/j.aim.2017.08.003 ) in several ways: (1) It does not require any bounds on second derivatives. (2) The covering number is replaced by the weaker notion of Gaussian-width. (3) We obtain stronger asymptotics with respect to the dimension. Two other corollaries are decomposition theorems for exponential random graphs and large-degree Ising models. In the Gaussian case, we show that measures of low-complexity exhibit an almost-tight reverse log-Sobolev inequality.