Moment Expansions of the Energy Distance
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Abstract (arXiv)
The energy distance is used to test distributional equality, and as a loss function in machine learning. While D2(X, Y)=0 only when X~ Y, the sensitivity to different moments is of practical importance. This work considers D2(X, Y) in the case where the distributions are close. In this regime, D2(X, Y) is more sensitive to differences in the means barX-barY, than differences in the covariances 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 Delta 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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References · 5
- 2016Optimization Methods for Large-Scale Machine Learningcited by 1,150
- 2023Neural General Circulation Models for Weather and Climatecited by 239
- 2017TensorFlow Distributionscited by 106
- 2021Probabilistic Forecasting with Generative Networks via Scoring Rule Minimizationcited by 37
- 2021Kernel Two-Sample Tests in High Dimension: Interplay Between Moment Discrepancy and Dimension-and-Sample Orderscited by 5
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