We study the tolerant testing problem for high-dimensional samplers. Given as input two samplers $\mathcal{P}$ and $\mathcal{Q}$ over the $n$-dimensional space $\{0,1\}^n$, and two parameters $\varepsilon_2 > \varepsilon_1$, the goal of tolerant testing is to test whether the distributions generated by $\mathcal{P}$ and $\mathcal{Q}$ are $\varepsilon_1$-close or $\varepsilon_2$-far. Since exponential lower bounds (in $n$) are known for the problem in the standard sampling model, research has focused on models where one can draw \textit{conditional} samples. Among these models, \textit{subcube conditioning} ($\mathsf{SUBCOND}$), which allows conditioning on arbitrary subcubes of the domain, holds the promise of widespread adoption in practice owing to its ability to capture the natural behavior of samplers in constrained domains. To translate the promise into practice, we need to overcome two crucial roadblocks for tests based on $\mathsf{SUBCOND}$: the prohibitively large number of queries ($\tilde{\mathcal{O}}(n^5/\varepsilon_2^5)$) and limitation to non-tolerant testing (i.e., $\varepsilon_1 = 0$). The primary contribution of this work is to overcome the above challenges: we design a new tolerant testing methodology (i.e., $\varepsilon_1 \geq 0$) that allows us to significantly improve the upper bound to $\tilde{\mathcal{O}}(n^3/(\varepsilon_2-\varepsilon_1)^5)$.

翻译：我们研究高维采样器的容忍性测试问题。给定两个在$n$维空间$\{0,1\}^n$上的采样器$\mathcal{P}$和$\mathcal{Q}$，以及两个参数$\varepsilon_2 > \varepsilon_1$，容忍性测试的目标是判断由$\mathcal{P}$和$\mathcal{Q}$生成的分布是$\varepsilon_1$-接近还是$\varepsilon_2$-远。由于在标准采样模型下该问题已知存在指数级下界（关于$n$），研究聚焦于可抽取\textit{条件}样本的模型。在这些模型中，\textit{子立方体条件}（$\mathsf{SUBCOND}$）允许对域中任意子立方体施加条件，因其能捕捉约束域中采样器的自然行为而有望在实践广泛应用。为将这一前景转化为实践，我们需要克服基于$\mathsf{SUBCOND}$的测试面临的两个关键障碍：查询量的过高数量级（$\tilde{\mathcal{O}}(n^5/\varepsilon_2^5)$）以及仅限于非容忍性测试（即$\varepsilon_1 = 0$）。本文的主要贡献在于克服上述挑战：我们设计了一种新的容忍性测试方法（即$\varepsilon_1 \geq 0$），使得上界显著改进至$\tilde{\mathcal{O}}(n^3/(\varepsilon_2-\varepsilon_1)^5)$。