The Geometry of Polynomial Group Convolutional Neural Networks

Yacoub Hendi; Daniel Persson; Magdalena Larfors

cs.LGMar 31, 2026

The Geometry of Polynomial Group Convolutional Neural Networks

Authors: Yacoub Hendi, Daniel Persson, Magdalena Larfors

Organizations: Uppsala University · Chalmers University of Technology

Abstract

We study polynomial group convolutional neural networks (PGCNNs) for an arbitrary finite group $G$ . In particular, we introduce a new mathematical framework for PGCNNs using the language of graded group algebras. This framework yields two natural parameterizations of the architecture, based on Hadamard and Kronecker products, related by a linear map. We compute the dimension of the associated neuromanifold, verifying that it depends only on the number of layers and the size of the group. Furthermore, we show the general fiber of both parameterizations is trivial up to the regular group action and rescaling. Hence both parametrization maps are identifiable.

Figures & tables

Figure 1: PGCNN parametrization maps $\varphi$ and $\Phi$ 2 2 2 The figure is generated by Gemini Pro. .

Panel of Figure 2 — Figure 2: A commutative diagram that describes the relation between the projectivized parametrization maps $\tilde{\Phi}$ and $\tilde{\varphi}$ induced by the PGCNN architecture with $L$ layers and activation degree $r$ over the finite group $G$ .

Group	$\mathbb{Z}_{2}$	$\mathbb{Z}_{3}$	$\mathbb{Z}_{4}$	$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$	$\mathbb{Z}_{5}$	$\mathbb{Z}_{6}$	$\mathbb{Z}_{2}\times\mathbb{Z}_{3}$	$S_{3}$
Max no. of layers	4	3	2	2	2	2	2	2

Table 1: Verification of Theorem 4.9 , using PGCNNGeometry , for PCGNN with the listed groups and maximum depth as shown in the second row; the activation degree is fixed to 2.

Explore similar work

Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks
Jun 1, 2026Michael AstwoodEquivariant Neural Networks Convolutional Neural Networks
Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2M6 Canada
Graph Neural Networks for Predicting Solvability of Finite Groups
May 30, 2026Tal WeissblatGraph Neural Networks
Expressivity of Shallow Neural Networks Over Finite Fields
Jul 19, 2026Maksym Zubkov, Carol Wu, Shiwei Yang +2Shallow Neural Networks Polynomial Neural Networks
Department of Mathematics, University of British Columbia, 1984 Mathematics Rd, Vancouver, V6T 1Z2, BC, Canada

Jun 1, 2026J/K move · Enter open · S save

Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks

Michael Astwood

Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2M6 Canada

Equivariant Neural Networks Convolutional Neural Networks

Open paper AskPDF arXiv

We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.

Abstract

Explore similar work

Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks

Graph Neural Networks for Predicting Solvability of Finite Groups

Expressivity of Shallow Neural Networks Over Finite Fields