Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.17772 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915564469878784 |
|---|---|
| author | Robinett, Ryan A. Madejski, Sophia A. Ruark, Kyle Riesenfeld, Samantha J. Orecchia, Lorenzo |
| author_facet | Robinett, Ryan A. Madejski, Sophia A. Ruark, Kyle Riesenfeld, Samantha J. Orecchia, Lorenzo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning Robinett, Ryan A. Madejski, Sophia A. Ruark, Kyle Riesenfeld, Samantha J. Orecchia, Lorenzo Machine Learning Applications I.5.1 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. |
| title | Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning |
| topic | Machine Learning Applications I.5.1 |
| url | https://arxiv.org/abs/2510.17772 |