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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.07072 |
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| _version_ | 1866914911804719104 |
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| author | Mataigne, Simon Absil, P. -A. Miolane, Nina |
| author_facet | Mataigne, Simon Absil, P. -A. Miolane, Nina |
| contents | We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by Hüper et al. (2021), which contains the well-known Euclidean and canonical metrics. First, we give the best Lipschitz constants between the distances induced by any two members of the family of metrics. Then, we give a lower and an upper bound on the geodesic distance by the easily computable Frobenius distance. We give explicit families of pairs of matrices that depend on the parameter of the metric and the dimensions of the manifold, where the lower and the upper bound are attained. These bounds aim at improving the theoretical guarantees and performance of minimal geodesic computation algorithms by reducing the initial velocity search space. In addition, these findings contribute to advancing the understanding of geodesic distances on the Stiefel manifold and their applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07072 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics Mataigne, Simon Absil, P. -A. Miolane, Nina Differential Geometry Machine Learning 15B10, 53Z50, 53C30, 14M17 We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by Hüper et al. (2021), which contains the well-known Euclidean and canonical metrics. First, we give the best Lipschitz constants between the distances induced by any two members of the family of metrics. Then, we give a lower and an upper bound on the geodesic distance by the easily computable Frobenius distance. We give explicit families of pairs of matrices that depend on the parameter of the metric and the dimensions of the manifold, where the lower and the upper bound are attained. These bounds aim at improving the theoretical guarantees and performance of minimal geodesic computation algorithms by reducing the initial velocity search space. In addition, these findings contribute to advancing the understanding of geodesic distances on the Stiefel manifold and their applications. |
| title | Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics |
| topic | Differential Geometry Machine Learning 15B10, 53Z50, 53C30, 14M17 |
| url | https://arxiv.org/abs/2408.07072 |