2023/09/26 by Weiming Feng, Liqiang Liu, Feng, Weiming +3 · 3 citations
Computer Science · #Algorithms and Data Compression #Bayesian Methods and Mixture Models #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning and Algorithms #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2309.14696
openalex publication_date 2023/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Total variation distance (TV distance) is an important measure for the difference between two distributions. Recently, there has been progress in approximating the TV distance between product distributions: a deterministic algorithm for a restricted class of product distributions (Bhattacharyya, Gayen, Meel, Myrisiotis, Pavan and Vinodchandran 2023) and a randomized algorithm for general product distributions (Feng, Guo, Jerrum and Wang 2023). We give a deterministic fully polynomial-time approximation algorithm (FPTAS) for the TV distance between product distributions. Given two product distributions ℙ and ℚ over [q]n, our algorithm approximates their TV distance with relative error ε in time O( (qn2)/(ε) log q log \fracnε ΔTV(ℙ,ℚ) ). Our algorithm is built around two key concepts: 1) The likelihood ratio as a distribution, which captures sufficient information to compute the TV distance. 2) We introduce a metric between likelihood ratio distributions, called the minimum total variation distance. Our algorithm computes a sparsified likelihood ratio distribution that is close to the original one w.r.t. the new metric. The approximated TV distance can be computed from the sparsified likelihood ratio. Our technique also implies deterministic FPTAS for the TV distance between Markov chains.