# Unsupervised Hierarchy Matching with Optimal Transport over Hyperbolic Spaces

arXiv: https://arxiv.org/abs/1911.02536 · HTML: https://mlsanity.com/paper/1911.02536/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Nov 2019.

## Built on by (most cited first, 14 of 14)

- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Using Optimal Transport as Alignment Objective for fine-tuning Multilingual Contextualized Embeddings](https://mlsanity.com/paper/2110.02887/) (2021)
- [Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals](https://mlsanity.com/paper/2303.05798/) (2023)
- [Riemannian Convex Potential Maps](https://mlsanity.com/paper/2106.10272/) (2021)
- [Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections](https://mlsanity.com/paper/2211.10066/) (2022)
- [Functional optimal transport: map estimation and domain adaptation for functional data](https://mlsanity.com/paper/2102.03895/) (2021)
- [Cycle Consistent Probability Divergences Across Different Spaces](https://mlsanity.com/paper/2111.11328/) (2021)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [Order Constraints in Optimal Transport](https://mlsanity.com/paper/2110.07275/) (2021)
- [Understanding in Artificial Intelligence](https://mlsanity.com/paper/2101.06573/) (2021)
- [Exploring Wasserstein Distance across Concept Embeddings for Ontology Matching](https://mlsanity.com/paper/2207.11324/) (2022)
- [Families of costs with zero and nonnegative MTW tensor in optimal transport and the c-divergences](https://mlsanity.com/paper/2401.00953/) (2024)
- [Sum-of-Squares Programming for Ma-Trudinger-Wang Regularity of Optimal Transport Maps](https://mlsanity.com/paper/2412.13372/) (2024)

## Abstract (arXiv)

This paper focuses on the problem of unsupervised alignment of hierarchical data such as ontologies or lexical databases. This is a problem that appears across areas, from natural language processing to bioinformatics, and is typically solved by appeal to outside knowledge bases and label-textual similarity. In contrast, we approach the problem from a purely geometric perspective: given only a vector-space representation of the items in the two hierarchies, we seek to infer correspondences across them. Our work derives from and interweaves hyperbolic-space representations for hierarchical data, on one hand, and unsupervised word-alignment methods, on the other. We first provide a set of negative results showing how and why Euclidean methods fail in this hyperbolic setting. We then propose a novel approach based on optimal transport over hyperbolic spaces, and show that it outperforms standard embedding alignment techniques in various experiments on cross-lingual WordNet alignment and ontology matching tasks.