Uniformity Testing over Hypergrids with Subcube Conditioning

TL;DR

This paper introduces an efficient algorithm for testing uniformity over hypergrids using subcube conditioning, extending previous work to more general domains with nearly optimal query complexity.

Contribution

The paper presents a new uniformity testing algorithm for hypergrids with improved query complexity and a novel Fourier-analytic proof of a robust Pisier's inequality.

Findings

Algorithm achieves nearly optimal query complexity for hypergrids.

Extends uniformity testing to general hypergrid domains.

Provides a new Fourier-analytic proof of a robust Pisier's inequality.

Abstract

We give an algorithm for testing uniformity of distributions supported on hypergrids $[m_1] \times \cdots \times [m_n]$, which makes $\smash{\widetilde{O}(\text{poly}(m)\sqrt{n}/\epsilon^2)}$ many queries to a subcube conditional sampling oracle with $m=\max_i m_i$. 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 $\{\pm 1\}^n$ 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.