Random Restrictions of High-Dimensional Distributions and Uniformity Testing with Subcube Conditioning

TL;DR

This paper introduces a nearly-optimal algorithm for uniformity testing of high-dimensional distributions using subcube conditioning, leveraging a novel random restriction technique and analyzing its impact on distribution means.

Contribution

It presents a new random restriction method for distributions on hypercubes and applies it to achieve optimal uniformity testing with subcube conditional samples.

Findings

Achieves nearly-optimal query complexity for uniformity testing.

Develops a natural notion of random restriction for high-dimensional distributions.

Provides a nearly-optimal mean testing algorithm with independent samples.

Abstract

We give a nearly-optimal algorithm for testing uniformity of distributions supported on $\{-1,1\}^n$, which makes $\tilde O (\sqrt{n}/\varepsilon^2)$ queries to a subcube conditional sampling oracle (Bhattacharyya and Chakraborty (2018)). The key technical component is a natural notion of random restriction for distributions on $\{-1,1\}^n$, and a quantitative analysis of how such a restriction affects the mean vector of the distribution. Along the way, we consider the problem of mean testing with independent samples and provide a nearly-optimal algorithm.