Tutorial material

Below you'll find a material for getting starting with differential geometry.

Weekend with Bernie!

This site collects introductory material on differential geometry for generative modeling. This site was born out of an ACML 2021 tutorial on the topic. Below you'll find slides, lecture notes, software and more. Feel free to get in contact with me in case you have thoughts on the topic.

In case you wonder why the site is called "Weekend with Bernie", then it is because the material is trying to be easily accessible (hence "weekend") and it leans on work by Bernhard Riemann (hence "Bernie"). The title may also be a reference to a fantastic movie...


Slides for the 10-part tutorial is available below. Click on a topic to get the corresponding slide deck; these are viewed in the browser and can be navigated with the arrow keys on your keyboard.

Part 1: Identifiability Part 2: Principal Component Analysis Part 3: Kery Idea
Part 4: Curve Lengths Part 5: Metrics Part 6: Geodesics
Part 7: Operational Representations Part 8: Noisy Manifolds Part 9: Case Studies
Part 10: The End

Lecture notes

The following notes are incomplete with many sections in a rough or incomplete state. The notes may nonetheless be valuable as existing material can be difficult to get into without a background in mathematics.

In case you have comments, then please do send them my way.

Download lecture notes


Practicalities can be a pain when working with differential geometry numerically. I won't pretend that we have solved all issues, but we have made it somewhat easy to get started. We have collected our most robust tools in the StochMan library (short for `stochastic manifolds'), which is available on Github.

Meet StochMan
StochMan software


The field is growing so I'm not always up to date with the latest references, but here are some relevant papers. This list is biased towards my own work as I know this the best; do feel free to share your work with me in case you think it should be on the list.

Original GP paper
Metrics for Probabilistic Geometries. A. Tosi, S. Hauberg, A. Vellido & ND. Lawrence. UAI 2014.
Random metrics in VAEs
Latent Space Oddity: on the Curvature of Deep Generative Models. G. Arvanitidis, LK. Hansen & S. Hauberg. ICLR 2018.
Deterministic metrics in VAEs
N. Chen, A. Klushyn, R. Kurle, X. Jiang, J. Bayer & P. Smagt. Metrics for deep generative models. AISTATS 2018
S. Laine. Feature-based metrics for exploring the latent space of generative models. ICLR Workshops 2018
H. Shao, A. Kumar & PT. Fletcher. The Riemannian geometry of deep generative models. CVPR Workshops 2018
On the importance of uncertainty
Only Bayes should learn a manifold. S. Hauberg. arxiv:1806.04994
Brownian motion priors
Variational Autoencoders with Riemannian Brownian Motion Priors. D. Kalatzis, D. Eklund, G. Arvanitidis & S. Hauberg. ICML 2020
Theoretical groundings
Expected path length on random manifolds. D. Eklund & S. Hauberg. arxiv:1908.07377
Alternative metrics
Geometrically enriched latent spaces. G. Arvanitidis, S. Hauberg & B. Schölkopf. AISTATS 2021
Dissimilarity data
Isometric Gaussian Process Latent Variable Model for Dissimilarity Data. M. Jørgensen & S. Hauberg. ICML 2021
Applications to protein data
What is a meaningful representation of protein sequences? NS. Detlefsen, S. Hauberg & W. Boomsma. arxiv:2012.02679
Applications to clustering
Geodesic Clustering in Deep Generative Models. T. Yang, G. Arvanitidis, D. Fu, X. Li & S. Hauberg. arxiv:1809.04747
Applications to robotics
Riemannian Manifold Learning for Robot Motion Skills. H. Beik-mohammadi, L. Rozo, G. Neumann & S. Hauberg. IROS workshop on geometry 2020
Learning Riemannian Manifolds for Geodesic Motion Skills. Hadi Beik-Mohammadi, Søren Hauberg, Georgios Arvanitidis, Gerhard Neumann and Leonel Rozo. Robotics: Science and Systems (RSS), 2021.
Trajectory Optimisation in Learned Multimodal Dynamical Systems Via Latent-ODE Collocation. A. Scannell, CH. Ek & A. Richards. ICRA 2021.
Active Learning based on Data Uncertainty and Model Sensitivity. N. Chen, A. Klushyn, A. Paraschos, D. Benbouzid, P. Smagt. arXiv:1808.02026, 2018
Non-Gaussian VAEs
Pulling back information geometry. G. Arvantidis, M. González-Duque, A. Pouplin, D. Kalatzis & S. Hauberg. arxiv:2106.05367, 2021.
Riemannian metric learning
Learning Riemannian Metrics. G. Lebanon. UAI 2012.
A Locally Adaptive Normal Distribution. G. Arvanitidis, L.K. Hansen & S. Hauberg. Neural Information Processing Systems (NeurIPS), 2016.
A Geometric Take on Metric Learning. S. Hauberg, O. Freifeld, and M.J. Black. In Advances in Neural Information Processing Systems (NeurIPS), 2012.