# Global Riemannian Acceleration in Hyperbolic and Spherical Spaces

arXiv: https://arxiv.org/abs/2012.03618 · HTML: https://mlsanity.com/paper/2012.03618/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2020.

## Built on by (most cited first, 10 of 10)

- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [Minimizing Dynamic Regret on Geodesic Metric Spaces](https://mlsanity.com/paper/2302.08652/) (2023)
- [Riemannian accelerated gradient methods via extrapolation](https://mlsanity.com/paper/2208.06619/) (2022)
- [Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties](https://mlsanity.com/paper/2211.14645/) (2022)
- [Non-Euclidean High-Order Smooth Convex Optimization](https://mlsanity.com/paper/2411.08987/) (2024)
- [Streamlining in the Riemannian Realm: Efficient Riemannian Optimization with Loopless Variance Reduction](https://mlsanity.com/paper/2403.06677/) (2024)
- [Smooth Quasar-Convex Optimization with Constraints](https://mlsanity.com/paper/2510.01943/) (2025)
- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)
- [Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions](https://mlsanity.com/paper/2607.20316/) (2026)
- [Retraction-Based Gradient Projection Algorithms on Manifolds](https://mlsanity.com/paper/2609.30885/) (2026)

## Abstract (arXiv)

We further research on the accelerated optimization phenomenon on Riemannian manifolds by introducing accelerated global first-order methods for the optimization of L-smooth and geodesically convex (g-convex) or μ-strongly g-convex functions defined on the hyperbolic space or a subset of the sphere. For a manifold other than the Euclidean space, these are the first methods to globally achieve the same rates as accelerated gradient descent in the Euclidean space with respect to L and ε (and μ if it applies), up to log factors. Due to the geometric deformations, our rates have an extra factor, depending on the initial distance R to a minimizer and the curvature K, with respect to Euclidean accelerated algorithms As a proxy for our solution, we solve a constrained non-convex Euclidean problem, under a condition between convexity and quasar-convexity, of independent interest. Additionally, for any Riemannian manifold of bounded sectional curvature, we provide reductions from optimization methods for smooth and g-convex functions to methods for smooth and strongly g-convex functions and vice versa. We also reduce global optimization to optimization over bounded balls where the effect of the curvature is reduced.