Geometric Clifford Algebra Networks

David Ruhe, Jayesh K Gupta, Steven De Keninck, Max Welling, Johannes Brandstetter
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:29306-29337, 2023.

Abstract

We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the $\mathrm{Pin}(p,q,r)$ group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-ruhe23a, title = {Geometric Clifford Algebra Networks}, author = {Ruhe, David and Gupta, Jayesh K and De Keninck, Steven and Welling, Max and Brandstetter, Johannes}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {29306--29337}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/ruhe23a/ruhe23a.pdf}, url = {https://proceedings.mlr.press/v202/ruhe23a.html}, abstract = {We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the $\mathrm{Pin}(p,q,r)$ group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.} }
Endnote
%0 Conference Paper %T Geometric Clifford Algebra Networks %A David Ruhe %A Jayesh K Gupta %A Steven De Keninck %A Max Welling %A Johannes Brandstetter %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-ruhe23a %I PMLR %P 29306--29337 %U https://proceedings.mlr.press/v202/ruhe23a.html %V 202 %X We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the $\mathrm{Pin}(p,q,r)$ group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.
APA
Ruhe, D., Gupta, J.K., De Keninck, S., Welling, M. & Brandstetter, J.. (2023). Geometric Clifford Algebra Networks. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:29306-29337 Available from https://proceedings.mlr.press/v202/ruhe23a.html.

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