We provide a unifying framework linking two classes of statistics used in
two-sample and independence testing: on the one hand, the energy distances and
distance covariances from the statistics literature; on the other, maximum mean
discrepancies (MMD), that is, distances between embeddings of distributions to
reproducing kernel Hilbert spaces (RKHS), as established in machine learning.
In the case where the energy distance is computed with a semimetric of negative
type, a positive definite kernel, termed distance kernel, may be defined such
that the MMD corresponds exactly to the energy distance. Conversely, for any
positive definite kernel, we can interpret the MMD as energy distance with
respect to some negative-type semimetric. This equivalence readily extends to
distance covariance using kernels on the product space. We determine the class
of probability distributions for which the test statistics are consistent
against all alternatives. Finally, we investigate the performance of the family
of distance kernels in two-sample and independence tests: we show in particular
that the energy distance most commonly employed in statistics is just one
member of a parametric family of kernels, and that other choices from this
family can yield more powerful tests.

本研究提供了一个统一的框架，将统计学文献中的能量距离和距离协方差与机器学习中的最大均值差（MMD）联系起来，并研究了这些统计量在一些概率分布下的可靠性和对多参数数据的适用性。

基于距离和 RKHS 统计量在假设检验中的等效性

Equivalence of distance-based and RKHS-based statistics in hypothesis  testing

We provide a unifying framework linking two classes of statistics used in
two-sample and independence testing: on the one hand, the energy distances and
distance covariances from the statistics literature; on the other, distances
between embeddings of distributions to reproducing kernel Hilbert spaces
(RKHS), as established in machine learning. The equivalence holds when energy
distances are computed with semimetrics of negative type, in which case a
kernel may be defined such that the RKHS distance between distributions
corresponds exactly to the energy distance. We determine the class of
probability distributions for which kernels induced by semimetrics are
characteristic (that is, for which embeddings of the distributions to an RKHS
are injective). Finally, we investigate the performance of this family of
kernels in two-sample and independence tests: we show in particular that the
energy distance most commonly employed in statistics is just one member of a
parametric family of kernels, and that other choices from this family can yield
more powerful tests.

本文提出了一个统一的框架，将统计领域中的能量距离和距离协方差与机器学习中的分布嵌入到重现核 Hilbert 空间之间的距离联系起来，并研究了半度量与其诱导的核函数的性质。最后，我们研究了这类核函数在两样本和独立性检验中的性能。