# Learning Embeddings into Entropic Wasserstein Spaces

arXiv: https://arxiv.org/abs/1905.03329 · HTML: https://mlsanity.com/paper/1905.03329/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2019.

## Built on by (most cited first, 15 of 15)

- [Identifiability Guarantees for Causal Disentanglement from Soft Interventions](https://mlsanity.com/paper/2307.06250/) (2023)
- [Towards Interpretable Deep Metric Learning with Structural Matching](https://mlsanity.com/paper/2108.05889/) (2021)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Wasserstein-Fisher-Rao Embedding: Logical Query Embeddings with Local Comparison and Global Transport](https://mlsanity.com/paper/2305.04034/) (2023)
- [Wasserstein Wormhole: Scalable Optimal Transport Distance with Transformers](https://mlsanity.com/paper/2404.09411/) (2024)
- [Re-evaluating Word Mover's Distance](https://mlsanity.com/paper/2105.14403/) (2021)
- [Neural Embedding: Learning the Embedding of the Manifold of Physics Data](https://mlsanity.com/paper/2208.05484/) (2022)
- [Learning to Generate Wasserstein Barycenters](https://mlsanity.com/paper/2102.12178/) (2021)
- [Wasserstein Graph Neural Networks for Graphs with Missing Attributes](https://mlsanity.com/paper/2102.03450/) (2021)
- [What is the Right Notion of Distance between Predict-then-Optimize Tasks?](https://mlsanity.com/paper/2409.06997/) (2024)
- [Selecting task with optimal transport self-supervised learning for few-shot classification](https://mlsanity.com/paper/2204.00289/) (2022)
- [Policy Search, Retrieval, and Composition via Task Similarity in Collaborative Agentic Systems](https://mlsanity.com/paper/2506.05577/) (2025)
- [Explaining Graph Neural Networks for Node Similarity on Graphs](https://mlsanity.com/paper/2407.07639/) (2024)
- [Wasserstein Hypergraph Neural Network](https://mlsanity.com/paper/2506.09682/) (2025)
- [Bispectral OT: Dataset Comparison using Symmetry-Aware Optimal Transport](https://mlsanity.com/paper/2509.20678/) (2025)

## Abstract (arXiv)

Euclidean embeddings of data are fundamentally limited in their ability to capture latent semantic structures, which need not conform to Euclidean spatial assumptions. Here we consider an alternative, which embeds data as discrete probability distributions in a Wasserstein space, endowed with an optimal transport metric. Wasserstein spaces are much larger and more flexible than Euclidean spaces, in that they can successfully embed a wider variety of metric structures. We exploit this flexibility by learning an embedding that captures semantic information in the Wasserstein distance between embedded distributions. We examine empirically the representational capacity of our learned Wasserstein embeddings, showing that they can embed a wide variety of metric structures with smaller distortion than an equivalent Euclidean embedding. We also investigate an application to word embedding, demonstrating a unique advantage of Wasserstein embeddings: We can visualize the high-dimensional embedding directly, since it is a probability distribution on a low-dimensional space. This obviates the need for dimensionality reduction techniques like t-SNE for visualization.