Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning

2025-10-20

AI Summary

Conventional manifold learning methods embed data into Euclidean space, often distorting intrinsic geometric and topological structure—especially when the embedding dimension approaches the true intrinsic dimension. Moreover, end-to-end learning of differentiable atlases remains unexplored. Method: We propose a general differentiable atlas data structure, enabling the first unsupervised, end-to-end manifold modeling directly on point clouds. Our approach performs representation learning and optimization on an underlying d-dimensional Riemannian manifold, integrating manifold-geometry-guided gradient propagation with Riemannian optimization to avoid distortion from Euclidean projection. Contribution/Results: Evaluated on Klein bottle classification and hematopoietic RNA velocity analysis, our method achieves superior accuracy, stability, and computational efficiency over standard embedding baselines. It demonstrates robustness to complex topology and fidelity to real biological data, significantly enhancing model interpretability and geometric faithfulness.

Technology Category

Machine Learning: Learning with ManifoldsComputer Vision: Representation Learning for VisionSearch and Optimization: Learning to Search

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsUser Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendationSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Developing differentiable atlas representations for Riemannian machine learning on manifolds
• Enabling direct optimization and learning on latent data manifolds
• Improving interpretability and robustness in manifold-based classification tasks

Innovation

• Differentiable atlas structure for Riemannian optimization
• Unsupervised heuristic learns atlas from point clouds
• Atlas-based manifold learning enhances interpretability and robustness