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Moment Expansions of the Energy Distance

Main:13 Pages
6 Figures
Bibliography:1 Pages
2 Tables
Appendix:4 Pages
Abstract

The energy distance is used to test distributional equality, and as a loss function in machine learning. While D2(X,Y)=0D^2(X, Y)=0 only when XYX\sim Y, the sensitivity to different moments is of practical importance. This work considers D2(X,Y)D^2(X, Y) in the case where the distributions are close. In this regime, D2(X,Y)D^2(X, Y) is more sensitive to differences in the means XˉYˉ\bar{X}-\bar{Y}, than differences in the covariances Δ\Delta. 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\Delta is examined when XX and YY 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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@article{langmore2025_2505.20647,
  title={ Moment Expansions of the Energy Distance },
  author={ Ian Langmore },
  journal={arXiv preprint arXiv:2505.20647},
  year={ 2025 }
}
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