Distance Estimation for High-Dimensional Discrete Distributions

TL;DR

This paper introduces the first polynomial-query algorithm for estimating the statistical distance between high-dimensional distributions using subcube conditional sampling, a model relevant for practical discrete data analysis.

Contribution

It presents a novel polynomial-query distance estimator in the subcube conditional sampling model, extending the capabilities of high-dimensional distribution testing.

Findings

Algorithm makes  (n^3/ ^5) queries

First polynomial-query estimator in model

Applicable to practical discrete sampling scenarios

Abstract

Given two distributions $\mathcal{P}$ and $\mathcal{Q}$ over a high-dimensional domain $\{0,1\}^n$, and a parameter $\varepsilon$, the goal of distance estimation is to determine the statistical distance between $\mathcal{P}$ and $\mathcal{Q}$, up to an additive tolerance $\pm \varepsilon$. Since exponential lower bounds (in $n$) are known for the problem in the standard sampling model, research has focused on richer query models where one can draw conditional samples. This paper presents the first polynomial query distance estimator in the conditional sampling model ($\mathsf{COND}$). We base our algorithm on the relatively weaker \textit{subcube conditional} sampling ($\mathsf{SUBCOND}$) oracle, which draws samples from the distribution conditioned on some of the dimensions. $\mathsf{SUBCOND}$ is a promising model for widespread practical use because it captures the natural behavior of discrete samplers. Our algorithm makes $\tilde{\mathcal{O}}(n^3/\varepsilon^5)$ queries to $\mathsf{SUBCOND}$.