We introduce a new version of the KL-divergence for Gaussian distributions which is based on Wasserstein geometry and referred to as WKL-divergence. We show that this version is consistent with the geometry of the sample space ${\Bbb R}^n$. In particular, we can evaluate the WKL-divergence of the Dirac measures concentrated in two points which turns out to be proportional to the squared distance between these points.

翻译：我们提出了一种基于Wasserstein几何的高斯分布KL散度的新版本，称为WKL散度。我们证明了该版本与样本空间${\Bbb R}^n$的几何结构具有一致性。特别地，我们可以评估集中在两个点上的狄拉克测度的WKL散度，结果表明该散度与这两点之间的平方距离成正比。