# Low-rank approximations of hyperbolic embeddings

## TL;DR

This paper introduces a method for learning hyperbolic embeddings constrained to low-rank subspaces, enabling efficient modeling of hierarchical data with shared structures.

## Contribution

It proposes a novel low-rank factorization approach for hyperbolic embeddings using manifold optimization techniques, which is computationally efficient and effective.

## Key findings

- Effective low-rank hyperbolic embeddings learned
- Improved modeling of hierarchical structures
- Efficient algorithms demonstrated

## Abstract

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.