Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows

Willem Diepeveen, Georgios Batzolis, Zakhar Shumaylov, Carola-Bibiane Schönlieb
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:13746-13773, 2025.

Abstract

Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient training algorithms. In this work, we integrate concepts from pullback Riemannian geometry and generative models to propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning: score-based pullback Riemannian geometry. Focusing on unimodal distributions as a first step, we propose a score-based Riemannian structure with closed-form geodesics that pass through the data probability density. With this structure, we construct a Riemannian autoencoder (RAE) with error bounds for discovering the correct data manifold dimension. This framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training. Through numerical experiments on diverse datasets, including image data, we demonstrate that the proposed framework produces high-quality geodesics passing through the data support, reliably estimates the intrinsic dimension of the data manifold, and provides a global chart of the manifold. To the best of our knowledge, this is the first scalable framework for extracting the complete geometry of the data manifold.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-diepeveen25a, title = {Score-based Pullback {R}iemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows}, author = {Diepeveen, Willem and Batzolis, Georgios and Shumaylov, Zakhar and Sch\"{o}nlieb, Carola-Bibiane}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {13746--13773}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/diepeveen25a/diepeveen25a.pdf}, url = {https://proceedings.mlr.press/v267/diepeveen25a.html}, abstract = {Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient training algorithms. In this work, we integrate concepts from pullback Riemannian geometry and generative models to propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning: score-based pullback Riemannian geometry. Focusing on unimodal distributions as a first step, we propose a score-based Riemannian structure with closed-form geodesics that pass through the data probability density. With this structure, we construct a Riemannian autoencoder (RAE) with error bounds for discovering the correct data manifold dimension. This framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training. Through numerical experiments on diverse datasets, including image data, we demonstrate that the proposed framework produces high-quality geodesics passing through the data support, reliably estimates the intrinsic dimension of the data manifold, and provides a global chart of the manifold. To the best of our knowledge, this is the first scalable framework for extracting the complete geometry of the data manifold.} }
Endnote
%0 Conference Paper %T Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows %A Willem Diepeveen %A Georgios Batzolis %A Zakhar Shumaylov %A Carola-Bibiane Schönlieb %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-diepeveen25a %I PMLR %P 13746--13773 %U https://proceedings.mlr.press/v267/diepeveen25a.html %V 267 %X Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient training algorithms. In this work, we integrate concepts from pullback Riemannian geometry and generative models to propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning: score-based pullback Riemannian geometry. Focusing on unimodal distributions as a first step, we propose a score-based Riemannian structure with closed-form geodesics that pass through the data probability density. With this structure, we construct a Riemannian autoencoder (RAE) with error bounds for discovering the correct data manifold dimension. This framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training. Through numerical experiments on diverse datasets, including image data, we demonstrate that the proposed framework produces high-quality geodesics passing through the data support, reliably estimates the intrinsic dimension of the data manifold, and provides a global chart of the manifold. To the best of our knowledge, this is the first scalable framework for extracting the complete geometry of the data manifold.
APA
Diepeveen, W., Batzolis, G., Shumaylov, Z. & Schönlieb, C.. (2025). Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:13746-13773 Available from https://proceedings.mlr.press/v267/diepeveen25a.html.

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