# Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning

arXiv: https://arxiv.org/abs/2510.17772 · HTML: https://mlsanity.com/paper/2510.17772/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 62 references that are in the map.
- First posted on arXiv 20 Oct 2025.

## Builds on (most cited first, 16 of 16)

- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [The non-convex Burer-Monteiro approach works on smooth semidefinite programs](https://mlsanity.com/paper/1606.04970/) (2016)
- [Fast and Eager k-Medoids Clustering: O(k) Runtime Improvement of the PAM, CLARA, and CLARANS Algorithms](https://mlsanity.com/paper/2008.05171/) (2020)
- [Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes](https://mlsanity.com/paper/2310.07216/) (2023)
- [An Alternative to EM for Gaussian Mixture Models: Batch and Stochastic Riemannian Optimization](https://mlsanity.com/paper/1706.03267/) (2017)
- [On the Burer-Monteiro method for general semidefinite programs](https://mlsanity.com/paper/1904.07147/) (2019)
- [Manifold Coordinates with Physical Meaning](https://mlsanity.com/paper/1811.11891/) (2018)
- [Principal Boundary on Riemannian Manifolds](https://mlsanity.com/paper/1711.06705/) (2017)
- [Efficient Diffusion Models for Symmetric Manifolds](https://mlsanity.com/paper/2505.21640/) (2025)

## Built on by (most cited first, 1 of 1)

- [Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases](https://mlsanity.com/paper/2602.22873/) (2026)

## Abstract (arXiv)

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 ℝD, 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.