New vector transport operators extending a Riemannian CG algorithm to generalized Stiefel manifold with low-rank applications
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911886494138368 |
|---|---|
| author | Wang, Xuejie Deng, Kangkang Peng, Zheng Yan, Chengcheng |
| author_facet | Wang, Xuejie Deng, Kangkang Peng, Zheng Yan, Chengcheng |
| contents | This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring-Wirth non-expansive condition under non-standard metrics, and one of them is also isometric. Building upon the novel vector transport operators, we extend the modified Polak-Ribi$\grave{e}$re-Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_00907 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New vector transport operators extending a Riemannian CG algorithm to generalized Stiefel manifold with low-rank applications Wang, Xuejie Deng, Kangkang Peng, Zheng Yan, Chengcheng Optimization and Control Numerical Analysis 90C26, 90C30, 90C15, 90C06, 90C90 This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring-Wirth non-expansive condition under non-standard metrics, and one of them is also isometric. Building upon the novel vector transport operators, we extend the modified Polak-Ribi$\grave{e}$re-Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis. |
| title | New vector transport operators extending a Riemannian CG algorithm to generalized Stiefel manifold with low-rank applications |
| topic | Optimization and Control Numerical Analysis 90C26, 90C30, 90C15, 90C06, 90C90 |
| url | https://arxiv.org/abs/2311.00907 |