# Aligning Hyperbolic Representations: an Optimal Transport-based approach

arXiv: https://arxiv.org/abs/2012.01089 · HTML: https://mlsanity.com/paper/2012.01089/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2020.

## Built on by (most cited first, 8 of 8)

- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [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)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [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)
- [Automated Hierarchical Graph Construction for Multi-source Electronic Health Records](https://mlsanity.com/paper/2509.06576/) (2025)

## Abstract (arXiv)

Hyperbolic-spaces are better suited to represent data with underlying hierarchical relationships, e.g., tree-like data. However, it is often necessary to incorporate, through alignment, different but related representations meaningfully. This aligning is an important class of machine learning problems, with applications as ontology matching and cross-lingual alignment. Optimal transport (OT)-based approaches are a natural choice to tackle the alignment problem as they aim to find a transformation of the source dataset to match a target dataset, subject to some distribution constraints. This work proposes a novel approach based on OT of embeddings on the Poincar'e model of hyperbolic spaces. Our method relies on the gyrobarycenter mapping on M"obius gyrovector spaces. As a result of this formalism, we derive extensions to some existing Euclidean methods of OT-based domain adaptation to their hyperbolic counterparts. Empirically, we show that both Euclidean and hyperbolic methods have similar performances in the context of retrieval.