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GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras

Ekaterina Filimoshina, Dmitry Shirokov

TL;DR

GLGENN tackles overparameterization in Clifford-algebra–based equivariant networks by introducing generalized Lipschitz groups and a parameter-light weight-sharing scheme. By operating on multivectors within $C \ell_{p,q,r}$ and enforcing equivariance via the $\tilde{\rm ad}$ action, GLGENN achieves pseudo-orthogonal equivariance with significantly fewer trainable parameters than prior GA-based models. Empirically, GLGENN matches or exceeds CGENN on regression, convex-hull-volume estimation, and N-body tasks across dimensions, while reducing overfitting in small-data regimes and improving training efficiency. The approach advances efficient, symmetry-aware neural architectures for scientific computing and offers a foundation for broader applications in physics-informed modeling and 3D geometric tasks.

Abstract

We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization technique that takes into account the fundamental structures and operations of geometric algebras. Due to this technique, GLGENN architecture is parameter-light and has less tendency to overfitting than baseline equivariant models. GLGENN outperforms or matches competitors on several benchmarking equivariant tasks, including estimation of an equivariant function and a convex hull experiment, while using significantly fewer optimizable parameters.

GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras

TL;DR

GLGENN tackles overparameterization in Clifford-algebra–based equivariant networks by introducing generalized Lipschitz groups and a parameter-light weight-sharing scheme. By operating on multivectors within and enforcing equivariance via the action, GLGENN achieves pseudo-orthogonal equivariance with significantly fewer trainable parameters than prior GA-based models. Empirically, GLGENN matches or exceeds CGENN on regression, convex-hull-volume estimation, and N-body tasks across dimensions, while reducing overfitting in small-data regimes and improving training efficiency. The approach advances efficient, symmetry-aware neural architectures for scientific computing and offers a foundation for broader applications in physics-informed modeling and 3D geometric tasks.

Abstract

We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization technique that takes into account the fundamental structures and operations of geometric algebras. Due to this technique, GLGENN architecture is parameter-light and has less tendency to overfitting than baseline equivariant models. GLGENN outperforms or matches competitors on several benchmarking equivariant tasks, including estimation of an equivariant function and a convex hull experiment, while using significantly fewer optimizable parameters.

Paper Structure

This paper contains 30 sections, 32 theorems, 156 equations, 6 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Geometric (Clifford) Algebras
  4. Lipschitz Groups and Twisted Adjoint Representations
  5. Equivariant Mappings and Neural Networks
  6. Theoretical Results
  7. Generalized Lipschitz Groups
  8. Pseudo-orthogonal Groups, Complex Orthogonal Groups, and Generalized Lipschitz Groups Equivariance
  9. Generalized and Ordinary Lipschitz Groups Equivariant Mappings
  10. Methodology
  11. Geometric Algebra Data Embeddings
  12. Conjugation Operations Layers
  13. $C \ell^{\overline{k}}_{p,q,r}$-Linear Layers
  14. $C \ell^{\overline{k}}_{p,q,r}$-Geometric Product Layers
  15. $C \ell^{\overline{k}}_{p,q,r}$-Normalization Layers
  16. ...and 15 more sections

Key Result

Lemma 2.1

Let $W\in C \ell^{\times}_{p,q,r}$, $U,V\in C \ell_{p,q,r}$, and $T\in(C \ell^{(0)\times}_{p,q,r}\cup C \ell^{(1)\times}_{p,q,r})\Lambda^{\times}_r$, where the notation $(C \ell^{(0)\times}_{p,q,r}\cup C \ell^{(1)\times}_{p,q,r})\Lambda^{\times}_r:=\{ab\;|\;a\in C \ell^{(0)\times}_{p,q,r}\cup for any $c\in C \ell^0$. Moreover, $\tilde{{\rm ad}}_T$ satisfies multiplicativity:

Figures (6)

  • Figure 1: GLGENN is an architecture of neural networks equivariant with respect to any pseudo-orthogonal transformation. Inputs and outputs are represented as multivectors (elements of geometric algebras), which encode various geometric quantities such as scalars, vectors, oriented areas (bivectors) and volumes (trivectors), and higher-dimensional objects (4-vectors, etc.). GLGENN are parameter-light, since they operate in a unified manner across $4$ fundamental subspaces of geometric algebras defined by the grade involution ($\widehat{\;\;}$) and reversion ($\widetilde{\;\;}$); they processes geometric objects in groups with a step size of $4$.
  • Figure 2: Left:${\rm O}(5,0)$-Regression. Middle:${\rm O}(5,0)$-Convex Hull, 16 points. The shaded regions depict 95% confidence intervals taken over 5 runs. Right:${\rm O}(5,0)$-$N$-Body. The shaded regions depict 95% confidence intervals taken over 3 runs.
  • Figure 3: Combination of MLP with GLGENN and CGENN in ${\rm O}(5,0)$-Regression.
  • Figure 4: ${\rm O}(5,0)$-Convex Hull, $K=16$. The plots illustrate the training and test loss curves for CGENN and GLGENN across different training iterations. Subfigures (A)–(D) correspond to different training set sizes: 256 (A), 1024 (B), 4096 (C), and 16384 (D), respectively.
  • Figure 5: ${\rm O}(5,0)$-Convex Hull, $K=256$. The plots illustrate the training and test loss curves for CGENN and GLGENN across different training iterations. Subfigures (A)–(B) correspond to training with $1024$ samples; subfigures (C)–(D) correspond to training with $16384$ samples. The right-hand plots zoom in on the final iterations of training.
  • ...and 1 more figures

Theorems & Definitions (67)

  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 57 more