On Approximating Total Variation Distance

Authors

Arnab Bhattacharyya, National University of Singapore
Sutanu Gayen, Indian Institute of Technology Kanpur
Kuldeep S. Meel, National University of Singapore
Dimitrios Myrisiotis, National University of Singapore
A. Pavan, Iowa State University
N. V. Vinodchandran, University of Nebraska-Lincoln

Document Type

Article

Date of this Version

2023

Citation

Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence (IJCAI-23)

Comments

Used by permission.

Abstract

Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the problem of computing the TV distance of two product distributions over the domain {0, 1}ⁿ. In particular, we establish the following results.

1. The problem of exactly computing the TV distance of two product distributions is #P-complete. This is in stark contrast with other distance measures such as KL, Chisquare, and Hellinger which tensorize over the marginals leading to efficient algorithms.
2. There is a fully polynomial-time deterministic approximation scheme (FPTAS) for computing the TV distance of two product distributions P and Q where Q is the uniform distribution. This result is extended to the case where Q has a constant number of distinct marginals. In contrast, we show that when P and Q are Bayes net distributions the relative approximation of their TV distance is NP-hard.