aiqbench analysis
This paper advances the theoretical foundations of machine learning by introducing a graded group algebra framework for polynomial group convolutional neural networks (PGCNNs). The authors rigorously analyze the architecture's parametrization via Hadamard and Kronecker products, deriving precise dimensions for the associated neuromanifold that depend solely on network depth and group size. These mathematical insights provide a deeper understanding of symmetry-based architectures, potentially enabling more efficient designs for invariant learning tasks.
The Geometry of Polynomial Group Convolutional Neural Networks
arXiv:2603.29566v2 Announce Type: replace 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 parametrizations 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. We also describe the general fiber of the Kronecker parametrization up to the regular group action and rescaling, and conjecture the analogous description for the Hadamard parametrization. Our conjecture is supported by explicit computations for small groups and shallow networks.
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 parametrizations 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. We also describe the general fiber of the Kronecker parametrization up to the regular group action and rescaling, and conjecture the analogous description for the Hadamard parametrization. Our conjecture is supported by explicit computations for small groups and shallow networks.
37 pages, Conjecture 4.8 in v1 is now proved as Proposition 4.8
Machine Learning (cs.LG); Algebraic Geometry (math.AG)
arXiv:2603.29566 [cs.LG]
(or arXiv:2603.29566v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2603.29566 Focus to learn more arXiv-issued DOI via DataCite
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