Uniformity Testing over Hypergrids with Subcube Conditioning
This addresses a fundamental problem in distribution testing for high-dimensional data, with incremental improvements over existing methods for hypergrids.
The paper tackles the problem of testing uniformity of distributions on hypergrids using a subcube conditional sampling oracle, achieving a query complexity of $\widetilde{O}( ext{poly}(m)\sqrt{n}/\epsilon^2)$, which is nearly optimal when $m$ is constant and improves upon prior work limited to hypercubes.
We give an algorithm for testing uniformity of distributions supported on hypergrids $[m_1] \times \cdots \times [m_n]$, which makes $\smash{\widetilde{O}(\text{poly}(m)\sqrt{n}/ε^2)}$ 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.