Curvature Estimation on Data Manifolds via Diffusion-augmented Sampling

Jason Wang, Bobak Kiani, Melanie Weber
Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, PMLR 282:736-769, 2026.

Abstract

Data geometry is fundamental to machine learning and data analysis, yet practical tools for characterizing the geometry of data manifolds remain limited. While intrinsic dimension estimation is well-studied, curvature, a key measure of local manifold structure, is far harder to approximate from noisy, sparsely sampled data. We introduce a diffusion-based framework for curvature estimation aiming to mitigate challenges due to low sample density. We train a diffusion model to learn a latent representation of the manifold, which we then probe to augment the raw dataset and obtain a denser sample. Compared to state-of-the-art curvature estimators applied directly to the raw data, diffusion-augmented methods achieve superior performance on heterogeneous manifolds when using high-fidelity diffusion models.

Cite this Paper

BibTeX
@InProceedings{pmlr-v282-wang26e, title = {Curvature Estimation on Data Manifolds via Diffusion-augmented Sampling}, author = {Wang, Jason and Kiani, Bobak and Weber, Melanie}, booktitle = {Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations}, pages = {736--769}, year = {2026}, editor = {Acosta, Francisco and Azeglio, Simone and Tolooshams, Bahareh and van de Geijn, Chase and Shewmake, Christian and Sanborn, Sophia and Miolane, Nina}, volume = {282}, series = {Proceedings of Machine Learning Research}, month = {14 Dec 2024--07 Dec 2025}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v282/main/assets/wang26e/wang26e.pdf}, url = {https://proceedings.mlr.press/v282/wang26e.html}, abstract = {Data geometry is fundamental to machine learning and data analysis, yet practical tools for characterizing the geometry of data manifolds remain limited. While intrinsic dimension estimation is well-studied, curvature, a key measure of local manifold structure, is far harder to approximate from noisy, sparsely sampled data. We introduce a diffusion-based framework for curvature estimation aiming to mitigate challenges due to low sample density. We train a diffusion model to learn a latent representation of the manifold, which we then probe to augment the raw dataset and obtain a denser sample. Compared to state-of-the-art curvature estimators applied directly to the raw data, diffusion-augmented methods achieve superior performance on heterogeneous manifolds when using high-fidelity diffusion models.} }
Endnote
%0 Conference Paper %T Curvature Estimation on Data Manifolds via Diffusion-augmented Sampling %A Jason Wang %A Bobak Kiani %A Melanie Weber %B Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations %C Proceedings of Machine Learning Research %D 2026 %E Francisco Acosta %E Simone Azeglio %E Bahareh Tolooshams %E Chase van de Geijn %E Christian Shewmake %E Sophia Sanborn %E Nina Miolane %F pmlr-v282-wang26e %I PMLR %P 736--769 %U https://proceedings.mlr.press/v282/wang26e.html %V 282 %X Data geometry is fundamental to machine learning and data analysis, yet practical tools for characterizing the geometry of data manifolds remain limited. While intrinsic dimension estimation is well-studied, curvature, a key measure of local manifold structure, is far harder to approximate from noisy, sparsely sampled data. We introduce a diffusion-based framework for curvature estimation aiming to mitigate challenges due to low sample density. We train a diffusion model to learn a latent representation of the manifold, which we then probe to augment the raw dataset and obtain a denser sample. Compared to state-of-the-art curvature estimators applied directly to the raw data, diffusion-augmented methods achieve superior performance on heterogeneous manifolds when using high-fidelity diffusion models.
APA
Wang, J., Kiani, B. & Weber, M.. (2026). Curvature Estimation on Data Manifolds via Diffusion-augmented Sampling. Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, in Proceedings of Machine Learning Research 282:736-769 Available from https://proceedings.mlr.press/v282/wang26e.html.

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