# Low-rank approximations of hyperbolic embeddings

arXiv: https://arxiv.org/abs/1903.07307 · HTML: https://mlsanity.com/paper/1903.07307/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Mar 2019.

## Built on by (most cited first, 3 of 3)

- [Riemannian accelerated gradient methods via extrapolation](https://mlsanity.com/paper/2208.06619/) (2022)
- [Riemannian coordinate descent algorithms on matrix manifolds](https://mlsanity.com/paper/2406.02225/) (2024)
- [Optimization over the intersection of manifolds](https://mlsanity.com/paper/2605.22736/) (2026)

## Abstract (arXiv)

The hyperbolic manifold is a smooth manifold of negative constant curvature. While the hyperbolic manifold is well-studied in the literature, it has gained interest in the machine learning and natural language processing communities lately due to its usefulness in modeling continuous hierarchies. Tasks with hierarchical structures are ubiquitous in those fields and there is a general interest to learning hyperbolic representations or embeddings of such tasks. Additionally, these embeddings of related tasks may also share a low-rank subspace. In this work, we propose to learn hyperbolic embeddings such that they also lie in a low-dimensional subspace. In particular, we consider the problem of learning a low-rank factorization of hyperbolic embeddings. We cast these problems as manifold optimization problems and propose computationally efficient algorithms. Empirical results illustrate the efficacy of the proposed approach.