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 φ and Φ 2 2 2 The figure is generated by Gemini Pro. .
Figure 2: A commutative diagram that describes the relation between the projectivized parametrization maps Φ~ and φ~ induced by the PGCNN architecture with L layers and activation degree r over the finite group G .
Group
Z2
Z3
Z4
Z2×Z2
Z5
Z6
Z2×Z3
S3
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.
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.
Michael Astwood
Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2M6 Canada
We present a Graph Neural Network (GNN) framework for the classification of finite groups according to their solvability. Using graph representations associated with finite groups, including Cayley graphs (CG), the proposed model is trained to distinguish solvable and non-solvable groups using structural graph information alone. The framework is evaluated on groups outside the training dataset in order to investigate the extent to which GNNs can learn algebraic properties arising in group theory. More broadly, the present work explores the relationship between algebraic structure and graph-based geometric representations of finite groups. The present study is intended as a proof-of-concept investigation of whether GNNs can learn algebraic properties of finite groups from graph-based representations
We study the expressivity of shallow polynomial neural networks (PNNs) with monomial activation functions over finite fields. For a given architecture, we define a neuromanifold as the image of the map from all possible network weights into the product of polynomial rings. We quantify the expressivity by the cardinality of the neuromanifold, and derive a natural lower and upper bound. This leads to counting rational points over finite fields, a problem closely linked to the Weil conjectures. Finally, we present an architecture that exhibits a striking difference in the neuromanifolds when considered over a characteristic zero versus a finite-characteristic field, illustrating the critical role of field characteristic in the notion of expressivity.
Maksym Zubkov, Carol Wu, Shiwei Yang +2
Department of Mathematics, University of British Columbia, 1984 Mathematics Rd, Vancouver, V6T 1Z2, BC, Canada