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 ±∈ the statistical (total variation) distance between their distributions. For some range of parameters, there is up to a logn 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 O(n logn).
ACM Transactions on Computation Theory
Watson, T. (2018). Communication complexity of statistical distance. ACM Transactions on Computation Theory, 10 (1) https://doi.org/10.1145/3170708