# Generalizing Point Embeddings using the Wasserstein Space of Elliptical Distributions

arXiv: https://arxiv.org/abs/1805.07594 · HTML: https://mlsanity.com/paper/1805.07594/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 May 2018.

## Built on by (most cited first, 20 of 20)

- [Multivariate Conformal Prediction using Optimal Transport](https://mlsanity.com/paper/2502.03609/) (2025)
- [Wasserstein-Fisher-Rao Embedding: Logical Query Embeddings with Local Comparison and Global Transport](https://mlsanity.com/paper/2305.04034/) (2023)
- [Duality of Bures and Shape Distances with Implications for Comparing Neural Representations](https://mlsanity.com/paper/2311.11436/) (2023)
- [Re-evaluating Word Mover's Distance](https://mlsanity.com/paper/2105.14403/) (2021)
- [Graph-Based Similarity of Neural Network Representations](https://mlsanity.com/paper/2111.11165/) (2021)
- [Critical Points and Convergence Analysis of Generative Deep Linear Networks Trained with Bures-Wasserstein Loss](https://mlsanity.com/paper/2303.03027/) (2023)
- [Concept2Box: Joint Geometric Embeddings for Learning Two-View Knowledge Graphs](https://mlsanity.com/paper/2307.01933/) (2023)
- [Personalised Federated Learning On Heterogeneous Feature Spaces](https://mlsanity.com/paper/2301.11447/) (2023)
- [Conditional Loss and Deep Euler Scheme for Time Series Generation](https://mlsanity.com/paper/2102.05313/) (2021)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [Representation Learning via Variational Bayesian Networks](https://mlsanity.com/paper/2306.16326/) (2023)
- [Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical models](https://mlsanity.com/paper/2403.10089/) (2024)
- [What is the Right Notion of Distance between Predict-then-Optimize Tasks?](https://mlsanity.com/paper/2409.06997/) (2024)
- [Towards Backwards-Compatible Data with Confounded Domain Adaptation](https://mlsanity.com/paper/2203.12720/) (2022)
- [Selecting task with optimal transport self-supervised learning for few-shot classification](https://mlsanity.com/paper/2204.00289/) (2022)
- [Robust probabilistic inference via a constrained transport metric](https://mlsanity.com/paper/2303.10085/) (2023)
- [UncertainGen: Uncertainty-Aware Representations of DNA Sequences for Metagenomic Binning](https://mlsanity.com/paper/2509.26116/) (2025)
- [Elliptical Ordinal Embedding](https://mlsanity.com/paper/2105.10457/) (2021)
- [GausSetExpander: A Simple Approach for Entity Set Expansion](https://mlsanity.com/paper/2202.13649/) (2022)
- [Bispectral OT: Dataset Comparison using Symmetry-Aware Optimal Transport](https://mlsanity.com/paper/2509.20678/) (2025)

## Abstract (arXiv)

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 a weighted 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 a 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 advantages of elliptical embeddings by using them for visualization, to compute embeddings of words, and to reflect entailment or hypernymy.