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

Abstract

We give a nearly-optimal algorithm for testing uniformity of distributions supported on {−1, 1} n , which makes O( √ n/ε 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.