Below you'll find a material for getting starting with differential geometry.
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.
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.
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
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|
|Expected path length on random manifolds. D. Eklund & S. Hauberg. arxiv:1908.07377|
|Geometrically enriched latent spaces. G. Arvanitidis, S. Hauberg & B. Schölkopf. AISTATS 2021|
|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|
|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.|