Papers › Gradient descent in hyperbolic space
Gradient descent in hyperbolic space
Benjamin Wilson, Matthias Leimeister
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
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.
In Syntology Open this paper in Syntology's Atlas, the map of the papers in Syntology's graph and their citations.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections