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## On Approximating Total Variation Distance

On Approximating Total Variation Distance.
Arnab Bhattacharyya, Sutanu Gayen, Kuldeep S Meel, Dimitrios Myrisiotis, Pavan A. and N. V. Vinodchandran.
In Proceedings of International Joint Conference on Artificial Intelligence (IJCAI), July 2023.

### 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, Chi-square, 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.

### BibTeX

```@inproceedings{BGMMAV23,
title={On Approximating Total Variation Distance},
author={  Bhattacharyya, Arnab and
Gayen, Sutanu and
Meel, Kuldeep S and
Myrisiotis, Dimitrios and
A., Pavan and
Vinodchandran, N. V.}
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, Chi-square, 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.}
publication_type={conference},
booktitle=IJCAI,
year={2023},
month=jul,
bib2html_rescat={Distribution Testing},
bib2html_pubtype={Refereed Conference},
bib2html_dl_pdf={../Papers/ijcai23-bggmpv.pdf},
}
```

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