mlsanityData to 7 Oct 2026

The Geometry of Polynomial Group Convolutional Neural Networks

arXiv 2603.29566 ↗First posted 31 Mar 2026cs.LG math.AGcited by 2

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Abstract (arXiv)

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.

Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 21 of its 35 references that are in the map. First posted on arXiv 31 Mar 2026.

Builds on

21 of 35 references.

References · 21

  1. 2016Group Equivariant Convolutional Networkscited by 1,154
  2. 2016Geometric deep learning: going beyond Euclidean datacited by 1,042
  3. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 223
  4. 2019On the Expressive Power of Deep Polynomial Neural Networkscited by 65
  5. 2021Geometric Deep Learning and Equivariant Neural Networkscited by 64
  6. 2020Deep Polynomial Neural Networkscited by 63
  7. 2017CryptoDL: Deep Neural Networks over Encrypted Datacited by 56
  8. 2019Pure and Spurious Critical Points: a Geometric Study of Linear Networkscited by 33
  9. 2021Geometry of Linear Convolutional Networkscited by 27
  10. 2025Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometrycited by 21
  11. 2018The loss surface of deep linear networks viewed through the algebraic geometry lenscited by 20
  12. 2018How Many Samples are Needed to Estimate a Convolutional or Recurrent Neural Network?cited by 19
9 more
  1. 2023Function Space and Critical Points of Linear Convolutional Networkscited by 19
  2. 2024Activation degree thresholds and expressiveness of polynomial neural networkscited by 14
  3. 2025Identifiability of Deep Polynomial Neural Networkscited by 12
  4. 2024On the Geometry and Optimization of Polynomial Convolutional Networkscited by 11
  5. 2024Algebraic Complexity and Neurovariety of Linear Convolutional Networkscited by 10
  6. 2025Learning on a Razor's Edge: Identifiability and Singularity of Polynomial Neural Networkscited by 9
  7. 2026Identifiable Equivariant Networks are Layerwise Equivariantcited by 7
  8. 2025The Alexander-Hirschowitz theorem for neurovarietiescited by 6
  9. 2025Equivariant non-linear maps for neural networks on homogeneous spacescited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026Conservation Laws from Data Symmetry in Neural Networkscited by 1
  2. 2026Singular Learning and Occam's Razor in Deep Monomial Networkscited by 0