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

Arnab Bhattacharyya, Sutanu Gayen, Kuldeep S. Meel, Dimitrios Myrisiotis, A. Pavan, N. V. Vinodchandran

TL;DR

This paper proves that the exact computation of the total variation distance between two product distributions is $\#\mathsf{P}$-complete, highlighting a sharp computational barrier for high-dimensional, succinctly described distributions. The authors establish membership in $\#\mathsf{P}$ via a nondeterministic counting approach and prove hardness through a reduction from $\#\textsc{PMFEquals}$, which itself reduces from $\#\textsc{SubsetProd}$; two case analyses depending on the value of $v$ are used to connect PMF-equal counts to TV-distance gaps. Unlike other divergences that tensorize over marginals and admit efficient computation, this result shows TV distance for product distributions inherits intrinsic counting complexity. The work situates TV distance within a broader landscape of hardness and approximation results, including SZK/NISZK hardness and recent FPRAS/FPTAS developments for related settings, underscoring a fundamental barrier to exact computation in high dimensions.

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.

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

TL;DR

This paper proves that the exact computation of the total variation distance between two product distributions is -complete, highlighting a sharp computational barrier for high-dimensional, succinctly described distributions. The authors establish membership in via a nondeterministic counting approach and prove hardness through a reduction from , which itself reduces from ; two case analyses depending on the value of are used to connect PMF-equal counts to TV-distance gaps. Unlike other divergences that tensorize over marginals and admit efficient computation, this result shows TV distance for product distributions inherits intrinsic counting complexity. The work situates TV distance within a broader landscape of hardness and approximation results, including SZK/NISZK hardness and recent FPRAS/FPTAS developments for related settings, underscoring a fundamental barrier to exact computation in high dimensions.

Abstract

We show that computing the total variation distance between two product distributions is -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.
Paper Structure (9 sections, 2 theorems, 7 equations)

This paper contains 9 sections, 2 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Notation:
  3. The Hardness of Computing TV Distance
  4. Membership in $\#\mathsf{P}$:
  5. $\#\mathsf{P}$-hardness:
  6. Case A: $v<2^{-n}$.
  7. Case B: $v \geq 2^{-n}$.
  8. Bibliographical Remarks
  9. Acknowledgements:

Key Result

Theorem 1

Given two product distributions $P$ and $Q$, computing $d_{\mathrm{TV}}(P,Q)$ is $\#\mathsf{P}$-complete.

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 2
  • proof
  • proof : Proof of \ref{['thm:hardness-Bern-products-intro']}
  • Claim 3
  • proof
  • Claim 4
  • proof
  • Claim 5
  • proof