# Horospherically Convex Optimization on Hadamard Manifolds Part I: Analysis and Algorithms

arXiv: https://arxiv.org/abs/2505.16970 · HTML: https://mlsanity.com/paper/2505.16970/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2025.

## Built on by (most cited first, 4 of 4)

- [Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives](https://mlsanity.com/paper/2509.11236/) (2025)
- [Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing](https://mlsanity.com/paper/2510.04579/) (2025)
- [1-Lipschitz Neural Networks on Hadamard Manifolds](https://mlsanity.com/paper/2607.19335/) (2026)
- [Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds](https://mlsanity.com/paper/2609.13646/) (2026)

## Abstract (arXiv)

Geodesic convexity (g-convexity) is a natural generalization of convexity to Riemannian manifolds. However, g-convexity lacks many desirable properties satisfied by Euclidean convexity. For instance, the natural notions of half-spaces and affine functions are themselves not g-convex. Moreover, recent studies have shown that the oracle complexity of geodesically convex optimization necessarily depends on the curvature of the manifold (Criscitiello and Boumal, 2022; Criscitiello and Boumal, 2023; Hamilton and Moitra, 2021), a computational bottleneck for several problems, e.g., tensor scaling. Recently, Lewis et al. (2024) addressed this challenge by proving curvature-independent convergence of subgradient descent, assuming horospherical convexity of the objective's sublevel sets. Using a similar idea, we introduce a generalization of convex functions to Hadamard manifolds, utilizing horoballs and Busemann functions as building blocks (as proxies for half-spaces and affine functions). We refer to this new notion as horospherical convexity (h-convexity). We provide algorithms for both nonsmooth and smooth h-convex optimization, which have curvature-independent guarantees exactly matching those from Euclidean space; this includes generalizations of subgradient descent and Nesterov's accelerated method. Motivated by applications, we extend these algorithms and their convergence rates to minimizing a sum of horospherically convex functions, assuming access to a weighted-Fr'echet-mean oracle.