Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning

Ryan Allen Robinett, Sophia Madejski, Kyle Ruark, Samantha J. Riesenfeld, Lorenzo Orecchia
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:3961-3969, 2026.

Abstract

Despite the popularity of the manifold hypothesis, current manifold-learning methods do not support machine learning directly on the latent $d$-dimensional data manifold, as they primarily aim to perform dimensionality reduction into $\mathbb{R}^D$, losing key manifold features when the embedding dimension $D$ approaches $d$. On the other hand, methods that directly learn the latent manifold as a differentiable atlas have been relatively underexplored. In this paper, we aim to give a proof of concept of the effectiveness and potential of atlas-based methods. To this end, we implement a generic data structure to maintain a differentiable atlas that enables Riemannian optimization over the manifold. We complement this with an unsupervised heuristic that learns a differentiable atlas from point cloud data. We experimentally demonstrate that this approach has advantages in terms of efficiency and accuracy in selected settings. Moreover, in a supervised classification task over the Klein bottle and in RNA velocity analysis of hematopoietic data, we showcase the improved interpretability and robustness of our approach.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-robinett26a, title = { Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning }, author = {Robinett, Ryan Allen and Madejski, Sophia and Ruark, Kyle and Riesenfeld, Samantha J. and Orecchia, Lorenzo}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {3961--3969}, 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/robinett26a/robinett26a.pdf}, url = {https://proceedings.mlr.press/v300/robinett26a.html}, abstract = { Despite the popularity of the manifold hypothesis, current manifold-learning methods do not support machine learning directly on the latent $d$-dimensional data manifold, as they primarily aim to perform dimensionality reduction into $\mathbb{R}^D$, losing key manifold features when the embedding dimension $D$ approaches $d$. On the other hand, methods that directly learn the latent manifold as a differentiable atlas have been relatively underexplored. In this paper, we aim to give a proof of concept of the effectiveness and potential of atlas-based methods. To this end, we implement a generic data structure to maintain a differentiable atlas that enables Riemannian optimization over the manifold. We complement this with an unsupervised heuristic that learns a differentiable atlas from point cloud data. We experimentally demonstrate that this approach has advantages in terms of efficiency and accuracy in selected settings. Moreover, in a supervised classification task over the Klein bottle and in RNA velocity analysis of hematopoietic data, we showcase the improved interpretability and robustness of our approach. } }
Endnote
%0 Conference Paper %T Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning %A Ryan Allen Robinett %A Sophia Madejski %A Kyle Ruark %A Samantha J. Riesenfeld %A Lorenzo Orecchia %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-robinett26a %I PMLR %P 3961--3969 %U https://proceedings.mlr.press/v300/robinett26a.html %V 300 %X Despite the popularity of the manifold hypothesis, current manifold-learning methods do not support machine learning directly on the latent $d$-dimensional data manifold, as they primarily aim to perform dimensionality reduction into $\mathbb{R}^D$, losing key manifold features when the embedding dimension $D$ approaches $d$. On the other hand, methods that directly learn the latent manifold as a differentiable atlas have been relatively underexplored. In this paper, we aim to give a proof of concept of the effectiveness and potential of atlas-based methods. To this end, we implement a generic data structure to maintain a differentiable atlas that enables Riemannian optimization over the manifold. We complement this with an unsupervised heuristic that learns a differentiable atlas from point cloud data. We experimentally demonstrate that this approach has advantages in terms of efficiency and accuracy in selected settings. Moreover, in a supervised classification task over the Klein bottle and in RNA velocity analysis of hematopoietic data, we showcase the improved interpretability and robustness of our approach.
APA
Robinett, R.A., Madejski, S., Ruark, K., Riesenfeld, S.J. & Orecchia, L.. (2026). Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:3961-3969 Available from https://proceedings.mlr.press/v300/robinett26a.html.

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