# Horoballs and the subgradient method

arXiv: https://arxiv.org/abs/2403.15749 · HTML: https://mlsanity.com/paper/2403.15749/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 26 references that are in the map.
- First posted on arXiv 23 Mar 2024.

## Builds on (most cited first, 6 of 6)

- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/2302.06807/) (2023)
- [Probabilistic smallest enclosing ball in high dimensions via subgradient sampling](https://mlsanity.com/paper/1902.10966/) (2019)
- [Convex analysis on Hadamard spaces and scaling problems](https://mlsanity.com/paper/2203.03193/) (2022)
- [A polynomial time algorithm to compute geodesics in CAT(0) cubical complexes](https://mlsanity.com/paper/1710.09932/) (2017)
- [CAT(0) geometry, robots, and society](https://mlsanity.com/paper/1912.10007/) (2019)

## Built on by (most cited first, 1 of 1)

- [Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives](https://mlsanity.com/paper/2509.11236/) (2025)

## Abstract (arXiv)

To explore convex optimization on Hadamard spaces, we consider an iteration in the style of a subgradient algorithm. Traditionally, such methods assume that the underlying spaces are manifolds and that the objectives are geodesically convex: the methods are described using tangent spaces and exponential maps. By contrast, our iteration applies in a general Hadamard space, is framed in the underlying space itself, and relies instead on horospherical convexity of the objective level sets. For this restricted class of objectives, we prove a complexity result of the usual form. Notably, the complexity does not depend on a lower bound on the space curvature. We illustrate our subgradient algorithm on the minimal enclosing ball problem in Hadamard spaces.