关于通过Zonotope压缩确定性计算全变差距离

中文摘要 AI 辅助

我们研究由简洁描述给出的高维分布之间的全变差距离的确定性相对近似。我们开发了一个抽象的确定性近似框架，该框架基于将全变差距离表示为低维zonotope的支撑函数。作为应用，我们为几个模型获得了FPTAS。给定两个在$[q]^n$上的乘积分布的混合，总共有$K$个分量分布，我们的算法在$\tilde O_K(nq(n/\varepsilon)^{2K})$时间内将其TV距离近似在$1+\varepsilon$的因子内。我们还给出了在$[q]^n$上具有总共$K$个分量分布的$n$步马尔可夫链混合的FPTAS，运行时间为$\tilde O_K(nq^2(n/\varepsilon)^{2K})$。最后，对于具有相同底层树拓扑的两个潜在树Ising模型，我们给出了它们叶边缘之间的TV距离的FPTAS，时间为$O(|V|^{13}\varepsilon^{-12})$。

英文摘要

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 $O(|V|^{13}\varepsilon^{-12})$.

发表机构

• The University of Hong Kong（香港大学）

机构由 AI 辅助整理，请以论文原文为准。