随机局部化的信息论观点
An Information-Theoretic View of Stochastic Localization
摘要：
给定${\mathbb R}^n$上的概率测度$μ$，通过少量概率测度的凸组合来近似它通常是有用的，使得每个分量接近于乘积测度。最近，Ronen Eldan使用随机局部化论证来证明这种类型的一般分解结果。在Eldan定理中，“成分的数量”由混合物的熵表征，“接近产品”由每个成分的协方差矩阵表征。我们提出了一个Eldan定理的初等证明，它利用了信息论（或估计论）的解释。证明类似于早期的分解结果之一，称为“钉扎引理”。'
英文摘要：
Given a probability measure $μ$ over ${\mathbb R}^n$, it is often useful to approximate it by the convex combination of a small number of probability measures, such that each component is close to a product measure. Recently, Ronen Eldan used a stochastic localization argument to prove a general decomposition result of this type. In Eldan's theorem, the `number of components' is characterized by the entropy of the mixture, and `closeness to product' is characterized by the covariance matrix of each component. We present an elementary proof of Eldan's theorem which makes use of an information theory (or estimation theory) interpretation. The proof is analogous to the one of an earlier decomposition result known as the `pinning lemma.'
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