On Computing Total Variation Distance Between Mixtures of Product Distributions

TL;DR

This paper presents algorithms for approximating the total variation distance between mixtures of product distributions, including a randomized approximation method and an exact deterministic algorithm for Boolean subcubes.

Contribution

It introduces a randomized approximation algorithm for general mixtures and a deterministic exact algorithm for Boolean subcube mixtures, analyzing computational hardness.

Findings

The randomized algorithm approximates total variation distance within (1±ε) in polynomial time.

The deterministic algorithm computes the distance exactly for Boolean subcubes in polynomial time.

Exact computation is -hard when the mixture components are linear in the dimension.

Abstract

We study the problem of approximating the total variation distance between two mixtures of product distributions over an $n$-dimensional discrete domain. Given two mixtures $\mathbb{P}$ and $\mathbb{Q}$ with $k_1$ and $k_2$ product distributions over $[q]^n$, respectively, we give a randomized algorithm that approximates $d_{\mathrm{TV}}\left({\mathbb{P}},{\mathbb{Q}}\right)$ within a multiplicative error of $(1\pm \varepsilon)$ in time $\mathrm{poly}((nq)^{k_1+k_2},1/\varepsilon)$. We also study the special case of mixtures of Boolean subcubes over $\{0,1\}^n$. For this class, we give a deterministic algorithm that exactly computes the total variation distance in time $\mathrm{poly}(n,2^{O(k_1+k_2)})$, and show that exact computation is $\#\mathsf{P}$-hard when $k_1+k_2=\Theta(n)$.