Total Variation Distance for Product Distributions is $\#\mathsf{P}$-Complete

TL;DR

This paper proves that calculating the total variation distance between two product distributions is computationally intractable (#P-complete), unlike other measures that allow efficient computation.

Contribution

It establishes the complexity of total variation distance for product distributions as #P-complete, highlighting a fundamental computational barrier.

Findings

Total variation distance computation is #P-complete.

Other measures like KL, Chi-square, Hellinger are efficiently computable for product distributions.

Abstract

We show that computing the total variation distance between two product distributions is $\#\mathsf{P}$-complete. This is in stark contrast with other distance measures such as Kullback-Leibler, Chi-square, and Hellinger, which tensorize over the marginals leading to efficient algorithms.