We give an algorithm for testing uniformity of distributions supported on hypergrids $[m]^n$, which makes $\tilde{O}(\text{poly}(m)\sqrt{n}/\epsilon^2)$ queries to a subcube conditional sampling oracle. When the side length $m$ of the hypergrid 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 $\mathbb{Z}_m^n$ using Fourier analysis.

翻译：我们提出一种算法，用于测试定义在超网格$[m]^n$上分布的均匀性。该算法使用子立方体条件采样（subcube conditional sampling）预言机，其查询复杂度为$\tilde{O}(\text{poly}(m)\sqrt{n}/\epsilon^2)$。当超网格边长$m$为常数时，该算法近乎最优，并强化了[CCK+21]的算法——后者虽具有相同查询复杂度，但仅适用于超立方体$\{\pm 1\}^n$。该算法分析中的关键技术贡献在于，利用傅里叶分析证明了关于$\mathbb{Z}_m^n$上函数的鲁棒版Pisier不等式。