Total Variation Distance Estimation through Domain Reduction

2026-09-16

AI Summary

本文通过领域缩减方法，特别是Lewis-weight采样技术，解决了高维分布总变差距离估计的难题，为混合积分布及一类结构化概率电路提供了一个多项式时间近似方案。

Abstract

Computing the total variation (TV) distance between succinctly represented high-dimensional distributions is generally intractable. We give an FPRAS for TV distance between mixtures of product distributions and, more generally, for a natural class of structured probabilistic circuits. Our main technique is a novel application of domain reduction: Given a family of feature vectors indexed by assignments, we use Lewis-weight sampling to replace the assignment domain by a polynomial-size weighted subset that simultaneously approximates the sum of absolute values of every linear projection. For mixtures of product distributions, we construct such reduced domains incrementally over the coordinates, obtaining the first FPRAS with running time polynomial in both the dimension and the number of mixture components. We then extend the approach to smooth, deterministic, structured-decomposable probabilistic circuits with a common structured architecture.

Problem

• Total Variation Distance
• High-dimensional Distributions
• Mixtures of Product Distributions
• Probabilistic Circuits

Innovation

• Total Variation Distance
• Domain Reduction
• Lewis-weight Sampling
• FPRAS
• Structured Probabilistic Circuits