# Only Bayes should learn a manifold (on the estimation of differential geometric structure from data)

arXiv: https://arxiv.org/abs/1806.04994 · HTML: https://mlsanity.com/paper/1806.04994/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2018.

## Built on by (most cited first, 17 of 17)

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- [The Geometry of Neural Nets' Parameter Spaces Under Reparametrization](https://mlsanity.com/paper/2302.07384/) (2023)
- [Addressing the Topological Defects of Disentanglement via Distributed Operators](https://mlsanity.com/paper/2102.05623/) (2021)
- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [Riemannian-Geometric Fingerprints of Generative Models](https://mlsanity.com/paper/2506.22802/) (2025)
- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Adversarial robustness of VAEs through the lens of local geometry](https://mlsanity.com/paper/2208.03923/) (2022)
- [Connecting Neural Models Latent Geometries with Relative Geodesic Representations](https://mlsanity.com/paper/2506.01599/) (2025)
- [Robust uncertainty estimates with out-of-distribution pseudo-inputs training](https://mlsanity.com/paper/2201.05890/) (2022)
- [Identifying latent distances with Finslerian geometry](https://mlsanity.com/paper/2212.10010/) (2022)
- [Learning Geometry and Topology via Multi-Chart Flows](https://mlsanity.com/paper/2505.24665/) (2025)
- [Geometry and Generalization: Eigenvalues as predictors of where a network will fail to generalize](https://mlsanity.com/paper/2107.06386/) (2021)
- [Geometric Contact Flows: Contactomorphisms for Dynamics and Control](https://mlsanity.com/paper/2506.17868/) (2025)
- [Geometric instability of out of distribution data across autoencoder architecture](https://mlsanity.com/paper/2201.11902/) (2022)
- [Interpolation of Missing Swaption Volatility Data using Gibbs Sampling on Variational Autoencoders](https://mlsanity.com/paper/2204.10400/) (2022)
- [A Likely Geometry of Generative Models](https://mlsanity.com/paper/2510.26266/) (2025)
- [Manifold Navigation Learning Using Recurrent Networks: Efficient Training Techniques](https://mlsanity.com/paper/2610.14325/) (2026)

## Abstract (arXiv)

We investigate learning of the differential geometric structure of a data manifold embedded in a high-dimensional Euclidean space. We first analyze kernel-based algorithms and show that under the usual regularizations, non-probabilistic methods cannot recover the differential geometric structure, but instead find mostly linear manifolds or spaces equipped with teleports. To properly learn the differential geometric structure, non-probabilistic methods must apply regularizations that enforce large gradients, which go against common wisdom. We repeat the analysis for probabilistic methods and find that under reasonable priors, the geometric structure can be recovered. Fully exploiting the recovered structure, however, requires the development of stochastic extensions to classic Riemannian geometry. We take early steps in that regard. Finally, we partly extend the analysis to modern models based on neural networks, thereby highlighting geometric and probabilistic shortcomings of current deep generative models.