Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning
This work addresses the challenge of preserving manifold features for interpretable Riemannian machine learning, offering a proof-of-concept for atlas-based methods in domains like computational biology.
The paper tackles the problem of enabling machine learning directly on latent data manifolds by introducing an atlas-based representation, demonstrating improved efficiency, accuracy, interpretability, and robustness in tasks like classification on the Klein bottle and RNA velocity analysis.
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.