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 $\mathsf{NP} \subseteq \mathsf{RP}$, it is impossible to efficiently estimate the TV distance between arbitrary Ising models, even in a bounded-error randomized setting.

翻译：我们研究了总变差（TV）距离中一些先前未被探索（或探索不足）的计算问题。首先，我们提出了一种简单的确定性多项式时间算法，用于检验任意字母表上乘积分布混合模型之间的等价性。这对应于一种特殊情况，即两个分布之间的总变差距离为零。其次，我们证明除非 $\mathsf{NP} \subseteq \mathsf{RP}$，否则即使在有界误差随机化设置下，也无法高效估计任意伊辛模型之间的总变差距离。