Computational Explorations of Total Variation Distance

We investigate some previously unexplored (or underexplored) computational aspects of total variation (TV) distance. First, we give a simple deterministic polynomial-time algorithm for checking equivalence between mixtures of product distributions, over arbitrary alphabets. This corresponds to a special case, whereby the TV distance between the two distributions is zero. Second, we prove that unless mathsfNP subseteq mathsfRP, it is impossible to efficiently estimate the TV distance between arbitrary Ising models, even in a bounded-error randomized setting.