Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres

Muskan Dosi, Chiranjeev Chiranjeev, Kartik Thakral, Mayank Vatsa, Richa Singh
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:14391-14416, 2025.

Abstract

Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Euclidean spaces. However, many real-world problems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to suboptimal generative performance. To address this limitation, we introduce HyperSphereDiff to align hyperspherical structures with directional noise, preserving class geometry and effectively capturing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic geometry of hyperspherical data, resulting in more accurate and geometry-aware generative models. We evaluate our framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underlying hyperspherical manifold.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-dosi25a, title = {Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres}, author = {Dosi, Muskan and Chiranjeev, Chiranjeev and Thakral, Kartik and Vatsa, Mayank and Singh, Richa}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {14391--14416}, 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/dosi25a/dosi25a.pdf}, url = {https://proceedings.mlr.press/v267/dosi25a.html}, abstract = {Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Euclidean spaces. However, many real-world problems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to suboptimal generative performance. To address this limitation, we introduce HyperSphereDiff to align hyperspherical structures with directional noise, preserving class geometry and effectively capturing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic geometry of hyperspherical data, resulting in more accurate and geometry-aware generative models. We evaluate our framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underlying hyperspherical manifold.} }
Endnote
%0 Conference Paper %T Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres %A Muskan Dosi %A Chiranjeev Chiranjeev %A Kartik Thakral %A Mayank Vatsa %A Richa Singh %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-dosi25a %I PMLR %P 14391--14416 %U https://proceedings.mlr.press/v267/dosi25a.html %V 267 %X Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Euclidean spaces. However, many real-world problems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to suboptimal generative performance. To address this limitation, we introduce HyperSphereDiff to align hyperspherical structures with directional noise, preserving class geometry and effectively capturing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic geometry of hyperspherical data, resulting in more accurate and geometry-aware generative models. We evaluate our framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underlying hyperspherical manifold.
APA
Dosi, M., Chiranjeev, C., Thakral, K., Vatsa, M. & Singh, R.. (2025). Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:14391-14416 Available from https://proceedings.mlr.press/v267/dosi25a.html.

Related Material