Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning

• URL: http://arxiv.org/abs/2510.17772v1
• Date: Mon, 20 Oct 2025 17:32:12 GMT
• Title: Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning
• Authors: Ryan A. Robinett, Sophia A. Madejski, Kyle Ruark, Samantha J. Riesenfeld, Lorenzo Orecchia,
• Abstract summary: We give a proof of concept of the effectiveness and potential of atlas-based methods.<n>We demonstrate that this approach has advantages in terms of efficiency in selected settings.<n>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.
• Score: 2.0155921857858474
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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