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

摘要

We give a nearly-optimal algorithm for testing uniformity of distributions supported on ? 1 1 n , which makes O ( n 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.