Horoballs and the subgradient method
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910397053796352 |
|---|---|
| author | Lewis, Adrian S. Lopez-Acedo, Genaro Nicolae, Adriana |
| author_facet | Lewis, Adrian S. Lopez-Acedo, Genaro Nicolae, Adriana |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15749 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Horoballs and the subgradient method Lewis, Adrian S. Lopez-Acedo, Genaro Nicolae, Adriana Optimization and Control Computational Complexity Machine Learning 90C48, 65Y20, 49M29 G.1.6 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. |
| title | Horoballs and the subgradient method |
| topic | Optimization and Control Computational Complexity Machine Learning 90C48, 65Y20, 49M29 G.1.6 |
| url | https://arxiv.org/abs/2403.15749 |