Random restrictions of high dimensional distributions and uniformity testing with subcube conditioning

We give a nearly-optimal algorithm for testing uniformity of distributions supported on {−1,1}n, which makes Oe(√n/ε2) many queries to a subcube conditional sampling oracle (Bhattacharyya and Chakraborty (2018)). The key technical component is a natural notion of random restrictions 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.