{"ID":2852913,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.17772","arxiv_id":"2510.17772","title":"Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning","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.","short_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 $D$ approach...","url_abs":"https://arxiv.org/abs/2510.17772","url_pdf":"https://arxiv.org/pdf/2510.17772v1","authors":"[\"Ryan A. Robinett\",\"Sophia A. Madejski\",\"Kyle Ruark\",\"Samantha J. Riesenfeld\",\"Lorenzo Orecchia\"]","published":"2025-10-20T17:32:12Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"stat.AP\"]","methods":"[]","has_code":false}