Generalized Lipschitz Group Equivariant Neural Networks (GLGENN)

Abstract

This repository contains implementation of Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to any pseudo-orthogonal transformation. The architecture of GLGENN contains

• ${C \kern -0.1em \ell}^{\overline{k}}_{p,q}$-linear layers,
• ${C \kern -0.1em \ell}^{\overline{k}}_{p,q}$-geometric product layers,
• ${C \kern -0.1em \ell}^{\overline{k}}_{p,q}$-normalization layers,

and employs theoretical results on generalized Lipschitz groups in Clifford algebras obtained in [1,2]. GLGENN generalize Clifford Group Equivariant Neural Networks presented in [3]

Code Organization

• algebra/: Contains implementation of quaternion types subspaces in Clifford algebra.
• data/: Contains data loading scripts for experiments.
• engineer/: Contains training, evaluation, and visualization scripts.
• experiments/: Contains experiments on GLGENN.
• layers/: Contains architecture of GLGENN layers.
• models/: Contains models built from GLGENN layers.

Experiments

• O(5) and O(7) regression tasks:
• Equivariance check:

References

[1] Filimoshina, E., Shirokov, D.: On generalization of Lipschitz groups and spin groups. Mathematical Methods in the Applied Sciences, 47(3), 1375--1400 (2024), arXiv:2205.06045
[2] Shirokov, D.: On inner automorphisms preserving fixed subspaces of Clifford algebras. Adv. Appl. Clifford Algebras 31(30), (2021), arXiv:2011.08287
[3] Ruhe, D., Brandstetter, J., Forré, P.: Clifford Group Equivariant Neural Networks (2023), arXiv:2305.11141