论文标题
用于高维两个样本问题的统一内核技巧
A uniform kernel trick for high-dimensional two-sample problems
论文作者
论文摘要
我们使用合适的版本的所谓“内核技巧”来设计两样本（同质性）测试，尤其是专注于高维和功能数据。我们的建议需要简化与选择适当的内核功能的重要实用问题有关的简化。具体而言，我们应用了核的统一变体，该变体涉及一类基于内核的距离内的the trick。我们获得了测试统计量的渐近分布（在零和替代假设下）。证明依赖于经验过程理论，结合了Delta方法和Hadamard（方向性）可区分性技术，以及基础过程的功能性Karhunen-Loève-type扩展。与文献中其他标准方法相比，该方法具有一些优势。与原始的基于内核的方法\ cite {gretton2007}相比，我们还对提案的性能和基于能量距离的测试\ cite \ cite {szekely-rizzo-2017}进行了一些实验见解。
We use a suitable version of the so-called "kernel trick" to devise two-sample (homogeneity) tests, especially focussed on high-dimensional and functional data. Our proposal entails a simplification related to the important practical problem of selecting an appropriate kernel function. Specifically, we apply a uniform variant of the kernel trick which involves the supremum within a class of kernel-based distances. We obtain the asymptotic distribution (under the null and alternative hypotheses) of the test statistic. The proofs rely on empirical processes theory, combined with the delta method and Hadamard (directional) differentiability techniques, and functional Karhunen-Loève-type expansions of the underlying processes. This methodology has some advantages over other standard approaches in the literature. We also give some experimental insight into the performance of our proposal compared to the original kernel-based approach \cite{Gretton2007} and the test based on energy distances \cite{Szekely-Rizzo-2017}.
