Moment Expansions of the Energy Distance
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916760367661056 |
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| author | Langmore, Ian |
| author_facet | Langmore, Ian |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_20647 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moment Expansions of the Energy Distance Langmore, Ian Machine Learning Statistics Theory 62G10 (Primary), 62E20, 62H15, 60E10 (Secondary) G.3 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. |
| title | Moment Expansions of the Energy Distance |
| topic | Machine Learning Statistics Theory 62G10 (Primary), 62E20, 62H15, 60E10 (Secondary) G.3 |
| url | https://arxiv.org/abs/2505.20647 |