Uniformity Testing over Hypergrids with Subcube Conditioning

We give an algorithm for testing uniformity of distributions supported on hypergrids [m1] × • • • × [mn], which makes O(poly(m) √ n/ϵ 2 ) many queries to a subcube conditional sampling oracle with m = maxi mi. When m is a constant, our algorithm is nearly optimal and strengthens the algorithm of [CCK + 21] which has the same query complexity but works for hypercubes ±1 n only.

A key technical contribution behind the analysis of our algorithm is a proof of a robust version of Pisier's inequality for functions over hypergrids using Fourier analysis.