# Gradient descent in hyperbolic space

arXiv: https://arxiv.org/abs/1805.08207 · HTML: https://mlsanity.com/paper/1805.08207/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2018.

## Built on by (most cited first, 11 of 11)

- [Lorentzian Graph Convolutional Networks](https://mlsanity.com/paper/2104.07477/) (2021)
- [The Numerical Stability of Hyperbolic Representation Learning](https://mlsanity.com/paper/2211.00181/) (2022)
- [Heterogeneous manifolds for curvature-aware graph embedding](https://mlsanity.com/paper/2202.01185/) (2022)
- [Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections](https://mlsanity.com/paper/2211.10066/) (2022)
- [Learning Weakly Supervised Audio-Visual Violence Detection in Hyperbolic Space](https://mlsanity.com/paper/2305.18797/) (2023)
- [Embedding Heterogeneous Networks into Hyperbolic Space Without Meta-path](https://mlsanity.com/paper/2106.09923/) (2021)
- [On Hyperbolic Embeddings in 2D Object Detection](https://mlsanity.com/paper/2203.08049/) (2022)
- [Learning phylogenetic trees as hyperbolic point configurations](https://mlsanity.com/paper/2104.11430/) (2021)
- [Stochastic Modified Flows for Riemannian Stochastic Gradient Descent](https://mlsanity.com/paper/2402.03467/) (2024)
- [A Geometry-Aware Algorithm to Learn Hierarchical Embeddings in Hyperbolic Space](https://mlsanity.com/paper/2407.16641/) (2024)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)

## Abstract (arXiv)

Gradient descent generalises naturally to Riemannian manifolds, and to hyperbolic n-space, in particular. Namely, having calculated the gradient at the point on the manifold representing the model parameters, the updated point is obtained by travelling along the geodesic passing in the direction of the gradient. Some recent works employing optimisation in hyperbolic space have not attempted this procedure, however, employing instead various approximations to avoid a calculation that was considered to be too complicated. In this tutorial, we demonstrate that in the hyperboloid model of hyperbolic space, the necessary calculations to perform gradient descent are in fact straight-forward. The advantages of the approach are then both illustrated and quantified for the optimisation problem of computing the Fr'echet mean (i.e. barycentre) of points in hyperbolic space.