On Deterministically Computing Total Variation Distance via Zonotope Compression

Abstract: We study deterministic relative approximation of the total variation distance between high-dimensional distributions given by succinct descriptions. We develop an abstract deterministic approximation framework based on representing the total variation distance as a support function of a low-dimensional zonotope. As applications, we obtain FPTASs for several models. Given two mixtures of product distributions over $[q]<sup>n$ with a total of $K$ component distributions, our algorithm approximates their TV-distance within a factor of $1+\varepsilon$ in time $\widetilde O_K(nq(n/\varepsilon)<sup>{2K})$. We also give an FPTAS for mixtures of $n$-step Markov chains over $[q]<sup>n$ with a total of $K$ component distributions, with running time $\widetilde O_K(nq<sup>2(n/\varepsilon)<sup>{2K})$. Finally, for two latent-tree Ising models with the same underlying tree topology, we give an FPTAS for the TV-distance between their leaf marginals in time $O(|V|<sup>{13}\varepsilon<sup>{-12})$.