On a Generalization of Wasserstein Distance and the Beckmann Problem to Connection Graphs
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910876217376768 |
|---|---|
| author | Robertson, Sawyer Kohli, Dhruv Mishne, Gal Cloninger, Alexander |
| author_facet | Robertson, Sawyer Kohli, Dhruv Mishne, Gal Cloninger, Alexander |
| contents | We propose a model of optimal parallel transport between vector fields on a connection graph, which consists of a weighted graph along with a map from its edges to an orthogonal group. Inspired by the well-known equivalence of 1-Wasserstein distance and minimum cost flows on standard graphs, we consider two versions of this problem: a minimum norm vector-valued flow problem with divergence constraints reflective of the connection structure of the graph; and a modified version which incorporates both quadratic regularization and a relaxation of the divergence constraint. Our theoretical contributions include: conditions for feasibility and computation of the Lagrangian dual problem for both problems, and duality correspondence for the relaxed-regularized version. Example applications of the model including transport between color images, vector field interpolation, and unsupervised clustering of vector field-valued data (in this case hurricane trajectory data) are also considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10295 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a Generalization of Wasserstein Distance and the Beckmann Problem to Connection Graphs Robertson, Sawyer Kohli, Dhruv Mishne, Gal Cloninger, Alexander Optimization and Control Discrete Mathematics 65K10, 05C21, 90C25, 68R10, 05C50 We propose a model of optimal parallel transport between vector fields on a connection graph, which consists of a weighted graph along with a map from its edges to an orthogonal group. Inspired by the well-known equivalence of 1-Wasserstein distance and minimum cost flows on standard graphs, we consider two versions of this problem: a minimum norm vector-valued flow problem with divergence constraints reflective of the connection structure of the graph; and a modified version which incorporates both quadratic regularization and a relaxation of the divergence constraint. Our theoretical contributions include: conditions for feasibility and computation of the Lagrangian dual problem for both problems, and duality correspondence for the relaxed-regularized version. Example applications of the model including transport between color images, vector field interpolation, and unsupervised clustering of vector field-valued data (in this case hurricane trajectory data) are also considered. |
| title | On a Generalization of Wasserstein Distance and the Beckmann Problem to Connection Graphs |
| topic | Optimization and Control Discrete Mathematics 65K10, 05C21, 90C25, 68R10, 05C50 |
| url | https://arxiv.org/abs/2312.10295 |