Distributed Simulation of Continuous Random Variables

Abstract: We establish the first known upper bound on the exact and Wyner's common information of $n$ continuous random variables in terms of the dual total correlation between them (which is a generalization of mutual information). In particular, we show that when the pdf of the random variables is log-concave, there is a constant gap of $n^{2}\log e+9n\log n$ between this upper bound and the dual total correlation lower bound that does not depend on the distribution. The upper bound is obtained using a computationally efficient dyadic decomposition scheme for constructing a discrete common randomness variable $W$ from which the $n$ random variables can be simulated in a distributed manner. We then bound the entropy of $W$ using a new measure, which we refer to as the erosion entropy.