分析能量距离对分布差异的敏感度，发现均值差比协方差差更显著。
Moment Expansions of the Energy Distance
- 通过展开能量距离，揭示其在分布接近时对均值差更敏感
- 协方差差的影响弱于均值差，且与维度无关
- 在各向同性近似下，非对角项贡献大幅降低
能量距离用于检验分布相等性，并作为机器学习中的损失函数。尽管 $D^2(X, Y)=0$ 仅当 $X\sim Y$ 时成立，但对不同矩的敏感度具有实际意义。本文研究分布接近时 $D^2(X, Y)$ 的性质。此时，$D^2(X, Y)$ 对均值差 $\bar{X}-\bar{Y}$ 的敏感度高于协方差差 $Δ$，这是由能量距离结构决定的，且与维度无关。当 $X$ 与 $Y$ 接近各向同性时，进一步考察 $Δ$ 的对角与非对角成分的敏感度。此时出现维度依赖的平均效应，多数情况下非对角相关性贡献显著减弱。数值实验验证了这些关系在分布假设不严格满足时依然成立。
原文摘要 · Abstract (English)
The energy distance is used to test distributional equality, and as a loss function in machine learning. While $D^2(X, Y)=0$ only when $X\sim Y$, the sensitivity to different moments is of practical importance. This work considers $D^2(X, Y)$ in the case where the distributions are close. In this regime, $D^2(X, Y)$ is more sensitive to differences in the means $\bar{X}-\bar{Y}$, than differences in the covariances $Δ$. This is due to the structure of the energy distance and is independent of dimension. The sensitivity to on versus off diagonal components of $Δ$ is examined when $X$ and $Y$ are close to isotropic. Here a dimension dependent averaging occurs and, in many cases, off diagonal correlations contribute significantly less. Numerical results verify these relationships hold even when distributional assumptions are not strictly met.
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