Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning

Ryan Allen Robinett, Sophia Madejski, Kyle Ruark, Samantha J. Riesenfeld, Lorenzo Orecchia

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.

Code Dataset Promise: Yes

Code Dataset Url: https://github.com/RiesenfeldGroup/atlas_graph_learning.git

Format Confirmation: I agree that I have read and followed the formatting instructions for the camera ready version.

Submission Number: 1712