Communication Complexity of Statistical Distance

We prove nearly matching upper and lower bounds on the randomized communication complexity of the following problem: Alice and Bob are each given a probability distribution over $n$ elements, and they wish to estimate within +-epsilon the statistical (total variation) distance between their distributions. For some range of parameters, there is up to a log(n) factor gap between the upper and lower bounds, and we identify a barrier to using information complexity techniques to improve the lower bound in this case. We also prove a side result that we discovered along the way: the randomized communication complexity of n-bit Majority composed with n-bit Greater-Than is Theta(n log n)