{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/parallel-transport-on-matrix-manifolds-and","title":"Parallel transport on matrix manifolds and Exponential Action","arxiv_id":"2408.06054","date":"2024-08-12","proceeding":null,"authors":["Du Nguyen","Stefan Sommer"],"abstract":"We express parallel transport for several common matrix Lie groups with a family of pseudo-Riemannian metrics in terms of matrix exponential and exponential actions. The expression for parallel transport is preserved by taking the quotient under certain scenarios. In particular, for a Stiefel manifold of orthogonal matrices of size $n\\times d$, we give an expression for parallel transport along a geodesic from time zero to $t$, that could be computed with time complexity of $O(nd^2)$ for small $t$, and of $O(td^3)$ for large t, contributing a step in a long-standing open problem in matrix manifolds. A similar result holds for flag manifolds with the canonical metric. We also show the parallel transport formulas for the generalized linear group, and the special orthogonal group under these metrics.","url_abs":"https://arxiv.org/abs/2408.06054v1","url_pdf":"https://arxiv.org/pdf/2408.06054v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"parallel-transport-on-matrix-manifolds-and","repo_url":"https://github.com/dnguyend/par-trans","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}