# Convex analysis on Hadamard spaces and scaling problems

arXiv: https://arxiv.org/abs/2203.03193 · HTML: https://mlsanity.com/paper/2203.03193/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Mar 2022.

## Built on by (most cited first, 2 of 2)

- [Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives](https://mlsanity.com/paper/2509.11236/) (2025)
- [Horoballs and the subgradient method](https://mlsanity.com/paper/2403.15749/) (2024)

## Abstract (arXiv)

In this paper, we address the bounded/unbounded determination of geodesically convex optimization on Hadamard spaces. In Euclidean convex optimization, the recession function is a basic tool to study the unboundedness, and provides the domain of the Legendre-Fenchel conjugate of the objective function. In a Hadamard space, the asymptotic slope function (Kapovich, Leeb, and Millson 2009), which is a function on the boundary at infinity, plays a role of the recession function. We extend this notion by means of convex analysis and optimization, and develop a convex analysis foundation for the unbounded determination of geodesically convex optimization on Hadamard spaces, particularly on symmetric spaces of nonpositive curvature. We explain how our developed theory is applied to operator scaling and related optimization on group orbits, which are our motivation.