Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets. We endow these measures with the 2-Wasserstein metric, with two important benefits: (i) For such measures, the squared 2-Wasserstein metric has a closed form, equal to the sum of the squared Euclidean distance between means and the squared Bures metric between covariance matrices. The latter is a Riemannian metric between positive semi-definite matrices, which turns out to be Euclidean on a suitable factor representation of such matrices, which is valid on the entire geodesic between these matrices. (ii) The 2-Wasserstein distance boils down to the usual Euclidean metric when comparing Diracs, and therefore provides the natural framework to extend point embeddings. We show that for these reasons Wasserstein elliptical embeddings are more intuitive and yield tools that are better behaved numerically than the alternative choice of Gaussian embeddings with the Kullback-Leibler divergence. In particular, and unlike previous work based on the KL geometry, we learn elliptical distributions that are not necessarily diagonal. We demonstrate the interest of elliptical embeddings by using them for visualization, to compute embeddings of words, and to reflect entanglement or hypernymy.

在机器学习中嵌入复杂对象是一个长期存在的问题。我们在这项工作中提出了一种方法，将对象作为椭圆概率分布嵌入。我们使用2-Wasserstein度量进行测量，并将它们用于词的嵌入和可视化。

利用椭球分布的Wasserstein空间泛化点嵌入