Learning Geometry and Topology via Multi-Chart Flows

Hanlin Yu, Søren Hauberg, Marcelo Hartmann, Arto Klami, Georgios Arvanitidis
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:2962-2970, 2026.

Abstract

Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if the manifold has a non-trivial topology, it can never be correctly learned using a single flow. Instead multiple flows must be ‘glued together’. In this paper, we first propose the general training scheme for learning such a collection of flows, and secondly we develop the first numerical algorithms for computing geodesics on such manifolds. Empirically, we demonstrate that this leads to highly significant improvements in topology estimation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-yu26c, title = { Learning Geometry and Topology via Multi-Chart Flows }, author = {Yu, Hanlin and Hauberg, S{\o}ren and Hartmann, Marcelo and Klami, Arto and Arvanitidis, Georgios}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {2962--2970}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/yu26c/yu26c.pdf}, url = {https://proceedings.mlr.press/v300/yu26c.html}, abstract = { Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if the manifold has a non-trivial topology, it can never be correctly learned using a single flow. Instead multiple flows must be ‘glued together’. In this paper, we first propose the general training scheme for learning such a collection of flows, and secondly we develop the first numerical algorithms for computing geodesics on such manifolds. Empirically, we demonstrate that this leads to highly significant improvements in topology estimation. } }
Endnote
%0 Conference Paper %T Learning Geometry and Topology via Multi-Chart Flows %A Hanlin Yu %A Søren Hauberg %A Marcelo Hartmann %A Arto Klami %A Georgios Arvanitidis %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-yu26c %I PMLR %P 2962--2970 %U https://proceedings.mlr.press/v300/yu26c.html %V 300 %X Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if the manifold has a non-trivial topology, it can never be correctly learned using a single flow. Instead multiple flows must be ‘glued together’. In this paper, we first propose the general training scheme for learning such a collection of flows, and secondly we develop the first numerical algorithms for computing geodesics on such manifolds. Empirically, we demonstrate that this leads to highly significant improvements in topology estimation.
APA
Yu, H., Hauberg, S., Hartmann, M., Klami, A. & Arvanitidis, G.. (2026). Learning Geometry and Topology via Multi-Chart Flows . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:2962-2970 Available from https://proceedings.mlr.press/v300/yu26c.html.

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