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.