Horoballs and the subgradient method
<|reference_start|>Horoballs and the subgradient method: 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.<|reference_end|>
@article{lewis2024horoballs,
    title={Horoballs and the subgradient method},
    author={Adrian S. Lewis and Genaro Lopez-Acedo and Adriana Nicolae},
    journal={arXiv preprint arXiv:2403.15749},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.15749},
    primaryClass={math.OC cs.CC cs.LG}
}