Distance correlation and Hilbert-Schmidt independence criterion are widely used for independence testing, two-sample testing, and many inference tasks in statistics and machine learning. These two methods are tightly related, yet are treated as two different entities in the majority of existing literature. In this paper, we propose a simple and elegant bijection between metric and kernel. The bijective transformation better preserves the similarity structure, allows distance correlation and Hilbert-Schmidt independence criterion to be always the same for hypothesis testing, streamlines the code base for implementation, and enables a rich literature of distance-based and kernel-based methodologies to directly communicate with each other.

距离相关性和Hilbert-Schmidt独立性准则被广泛用于独立性测试，两样本测试以及统计和机器学习中的许多推理任务。这两种方法紧密相关，但在大多数现有文献中被视为两个不同的实体。在本文中，我们提出了度量与内核之间简单而优雅的双射。双射变换更好地保留了相似性结构，允许距离相关性和希尔伯特-施密特独立性准则在假设检验中始终相同，简化了实现的代码库，并使基于距离和基于内核的方法论的丰富文献得以直接互相交流。