Common Information Dimension
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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 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,⋯,X_n using a member of ℱ. Our main contributions include the computation of the common information dimension for jointly Gaussian random vectors in a closed form, with ℱ being the linear functions class.
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