# The Geometry of Polynomial Group Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2603.29566 · HTML: https://mlsanity.com/paper/2603.29566/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 35 references that are in the map.
- First posted on arXiv 31 Mar 2026.

## Builds on (most cited first, 21 of 21)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [On the Expressive Power of Deep Polynomial Neural Networks](https://mlsanity.com/paper/1905.12207/) (2019)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Deep Polynomial Neural Networks](https://mlsanity.com/paper/2006.13026/) (2020)
- [CryptoDL: Deep Neural Networks over Encrypted Data](https://mlsanity.com/paper/1711.05189/) (2017)
- [Pure and Spurious Critical Points: a Geometric Study of Linear Networks](https://mlsanity.com/paper/1910.01671/) (2019)
- [Geometry of Linear Convolutional Networks](https://mlsanity.com/paper/2108.01538/) (2021)
- [Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry](https://mlsanity.com/paper/2501.18915/) (2025)
- [The loss surface of deep linear networks viewed through the algebraic geometry lens](https://mlsanity.com/paper/1810.07716/) (2018)
- [How Many Samples are Needed to Estimate a Convolutional or Recurrent Neural Network?](https://mlsanity.com/paper/1805.07883/) (2018)
- [Function Space and Critical Points of Linear Convolutional Networks](https://mlsanity.com/paper/2304.05752/) (2023)
- [Activation degree thresholds and expressiveness of polynomial neural networks](https://mlsanity.com/paper/2408.04569/) (2024)
- [Identifiability of Deep Polynomial Neural Networks](https://mlsanity.com/paper/2506.17093/) (2025)
- [On the Geometry and Optimization of Polynomial Convolutional Networks](https://mlsanity.com/paper/2410.00722/) (2024)
- [Algebraic Complexity and Neurovariety of Linear Convolutional Networks](https://mlsanity.com/paper/2401.16613/) (2024)
- [Learning on a Razor's Edge: Identifiability and Singularity of Polynomial Neural Networks](https://mlsanity.com/paper/2505.11846/) (2025)
- [Identifiable Equivariant Networks are Layerwise Equivariant](https://mlsanity.com/paper/2601.21645/) (2026)
- [The Alexander-Hirschowitz theorem for neurovarieties](https://mlsanity.com/paper/2511.19703/) (2025)
- [Equivariant non-linear maps for neural networks on homogeneous spaces](https://mlsanity.com/paper/2504.20974/) (2025)

## Built on by (most cited first, 2 of 2)

- [Conservation Laws from Data Symmetry in Neural Networks](https://mlsanity.com/paper/2606.10913/) (2026)
- [Singular Learning and Occam's Razor in Deep Monomial Networks](https://mlsanity.com/paper/2606.28464/) (2026)

## 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.