Moment Expansions of the Energy Distance
This provides theoretical insights for practitioners using the energy distance in statistical testing and machine learning, but it is incremental as it builds on existing properties without introducing new methods.
The paper analyzes the sensitivity of the energy distance to different moments when distributions are close, finding it is more sensitive to mean differences than covariance differences, with off-diagonal covariance components contributing less in isotropic cases.
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.