Total correlation (`TC') and dual total correlation (`DTC') are two classical ways to quantify the correlation among an $n$-tuple of random variables. They both reduce to mutual information when $n=2$. The first part of this paper sets up the theory of TC and DTC for general random variables, not necessarily finite-valued. This generality has not been exposed in the literature before. The second part considers the structural implications when a joint distribution $\mu$ has small TC or DTC. If $\mathrm{TC}(\mu) = o(n)$, then $\mu$ is close to a product measure according to a suitable transportation metric: this follows directly from Marton's classical transportation-entropy inequality. If $\mathrm{DTC}(\mu) = o(n)$, then the structural consequence is more complicated: $\mu$ is a mixture of a controlled number of terms, most of them close to product measures in the transportation metric. This is the main new result of the paper.

中文翻译：

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多变量相关性和产品度量的混合

全相关（`TC'）和对偶全相关（`DTC'）是量化$n$-随机变量元组之间相关性的两种经典方法。当 $n=2$ 时，它们都简化为互信息。本文的第一部分建立了一般随机变量的 TC 和 DTC 理论，不一定是有限值的。这种普遍性以前没有在文献中公开过。第二部分考虑了当联合分布 $\mu$ 具有小 TC 或 DTC 时的结构含义。如果 $\mathrm{TC}(\mu) = o(n)$，则 $\mu$ 接近于根据合适的运输度量的产品度量：这直接来自 Marton 的经典运输-熵不等式。如果 $\mathrm{DTC}(\mu) = o(n)$，则结构结果更复杂：$\mu$ 是受控数量项的混合，它们中的大多数接近运输指标中的产品度量。这是论文的主要新成果。