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Computational Explorations of Total Variation Distance

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

TL;DR

The paper tackles computational questions about total variation distance, proving a deterministic polynomial-time algorithm for exact equivalence checking of mixtures of product distributions and establishing hardness results for Ising models. The equivalence algorithm uses a vector-space basis construction to efficiently confirm equality and provide a witness when they differ, with running time $O(n k^4 |\Sigma|^4)$. Conversely, it shows that, unless $\mathsf{NP} \subseteq \mathsf{RP}$, there is no FPRAS for estimating $d_{\mathrm{TV}}$ between Ising models, via a sequence of reductions from the partition function to atomic marginals and then to TV distance. The results illuminate fundamental computational limits of TV-distance tasks in structured distributions and suggest cautious optimism for efficient equivalence testing in mixtures of product distributions, while highlighting hardness barriers in broader graphical models.

Abstract

We investigate some previously unexplored (or underexplored) computational aspects of total variation (TV) distance. First, we give a simple deterministic polynomial-time algorithm for checking equivalence between mixtures of product distributions, over arbitrary alphabets. This corresponds to a special case, whereby the TV distance between the two distributions is zero. Second, we prove that unless $\mathsf{NP} \subseteq \mathsf{RP}$, it is impossible to efficiently estimate the TV distance between arbitrary Ising models, even in a bounded-error randomized setting.

Computational Explorations of Total Variation Distance

TL;DR

The paper tackles computational questions about total variation distance, proving a deterministic polynomial-time algorithm for exact equivalence checking of mixtures of product distributions and establishing hardness results for Ising models. The equivalence algorithm uses a vector-space basis construction to efficiently confirm equality and provide a witness when they differ, with running time . Conversely, it shows that, unless , there is no FPRAS for estimating between Ising models, via a sequence of reductions from the partition function to atomic marginals and then to TV distance. The results illuminate fundamental computational limits of TV-distance tasks in structured distributions and suggest cautious optimism for efficient equivalence testing in mixtures of product distributions, while highlighting hardness barriers in broader graphical models.

Abstract

We investigate some previously unexplored (or underexplored) computational aspects of total variation (TV) distance. First, we give a simple deterministic polynomial-time algorithm for checking equivalence between mixtures of product distributions, over arbitrary alphabets. This corresponds to a special case, whereby the TV distance between the two distributions is zero. Second, we prove that unless , it is impossible to efficiently estimate the TV distance between arbitrary Ising models, even in a bounded-error randomized setting.

Paper Structure

This paper contains 15 sections, 6 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Equivalence Checking for Mixtures of Product Distributions
  3. Hardness of Approximating Total Variation Distance Between Ising Models
  4. Paper Organization
  5. Preliminaries
  6. Equivalence Checking for Mixtures of Product Distributions
  7. Base Case.
  8. Induction Hypothesis.
  9. Induction Step.
  10. Running Time.
  11. Hardness of Approximating Total Variation Distance Between Ising Models
  12. Reducing the Partition Function to Atomic Marginals
  13. Reducing the Atomic Marginals to TV Distance
  14. Proof of
  15. Discussion

Key Result

Theorem 1

There is a deterministic algorithm $E$ such that, given two mixtures of product distributions $P$ and $Q$, specified by $(w_1, \dots, w_k, P_1, \dots, P_k)$ and $(v_1, \dots, v_k, Q_1, \dots, Q_k)$, respectively, decides whether $P = Q$ or not. Moreover, if $P \neq Q$, then $E$ outputs some $x \in \

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • proof
  • Definition 5
  • Claim 6
  • proof : Proof of \ref{['clm:basis-products']}
  • Proposition 7
  • proof
  • Proposition 8
  • ...and 3 more