The exact common information between a set of random variables $X_1,...,X_n$ is defined as the minimum entropy of a shared random variable that allows for the exact distributive simulation of $X_1,...,X_n$. It has been established that, in certain instances, infinite entropy is required to achieve distributive simulation, suggesting that continuous random variables may be needed in such scenarios. However, to date, there is no established metric to characterize such cases. In this paper, we propose the concept of Common Information Dimension (CID) with respect to a given class of functions $\mathcal{F}$, defined as the minimum dimension of a random variable $W$ required to distributively simulate a set of random variables $X_1,...,X_n$, such that $W$ can be expressed as a function of $X_1,\cdots,X_n$ using a member of $\mathcal{F}$. Our main contributions include the computation of the common information dimension for jointly Gaussian random vectors in a closed form, with $\mathcal{F}$ being the linear functions class.

翻译：一组随机变量 $X_1,...,X_n$ 的精确公共信息定义为：允许对这些随机变量进行精确分布式模拟的共享随机变量的最小熵。已有研究表明，在某些情况下，实现分布式模拟需要无限熵，这表明此类场景可能需要连续随机变量。然而，迄今为止尚未建立用于刻画此类情形的度量标准。本文针对给定函数类 $\mathcal{F}$ 提出"公共信息维度（CID）"概念，将其定义为进行分布式模拟 $X_1,...,X_n$ 所需的最小随机变量 $W$ 的维度，且该 $W$ 可通过 $\mathcal{F}$ 中的某个函数表示为 $X_1,\cdots,X_n$ 的函数。我们的主要贡献包括：在 $\mathcal{F}$ 为线性函数类时，以闭式形式计算联合高斯随机向量的公共信息维度。