🤖 AI Summary
本文通过将总变差距离表示为低维zonotope的支撑函数，开发了一个抽象的确定性近似框架，解决了高维分布间总变差距离的确定性相对近似问题。
📝 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 $[q]^n$ with a total of $K$ component distributions, our algorithm approximates their TV-distance within a factor of $1+\varepsilon$ in time $\widetilde O_K(nq(n/\varepsilon)^{2K})$. We also give an FPTAS for mixtures of $n$-step Markov chains over $[q]^n$ with a total of $K$ component distributions, with running time $\widetilde O_K(nq^2(n/\varepsilon)^{2K})$. 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 $O(|V|^{13}\varepsilon^{-12})$.