PGCNNGeometry

PGCNN

Polynomial Group-Convolutional Neural Networks (PGCNN)

A Python/Sage package for algebraic analysis of Polynomial Group-Convolutional Neural Networks, including:

• symbolic computation of network parametrization maps
• image dimension computation
• generic fiber size computation
• algebraic geometry analysis via Macaulay2

The package integrates:

• SageMath for symbolic computation and group algebra
• Macaulay2 (with the Cremona package) for algebraic geometry computations.

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Requirements

The package requires the following software:

• Python ≥ 3.9
• SageMath
• Macaulay2
• Macaulay2 package Cremona

Because SageMath is not a standard pip dependency, this package must be installed using Sage’s Python environment.

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Installation

1. Install SageMath

Follow the instructions for your system:

https://doc.sagemath.org/html/en/installation/index.html

Verify installation:

sage --version

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2. Install Macaulay2

Instructions:

https://macaulay2.com/

Verify installation:

M2

Inside Macaulay2 install the Cremona package:

installPackage "Cremona"

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3. Install the PGCNN package

sage -python -m pip install pgcnn

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Verify Installation

Start Sage Python:

sage -python

Run the following test:

from sage.all import CyclicPermutationGroup
from pgcnn import PGCNN

G = CyclicPermutationGroup(2)

net = PGCNN(G, [1])

print(net.compute_dim_image_Phi_sage())

If this runs without errors, the installation is successful.

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Example Usage

from sage.all import CyclicPermutationGroup
from pgcnn import PGCNN

# Define a group
G = CyclicPermutationGroup(3)

# Define activation exponents
r = [1, 2]

# Create network
net = PGCNN(G, r)

# Compute image dimension
dim = net.compute_dim_image_Phi_sage()

print("Image dimension:", dim)

See example.sage for more examples.

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Citation

If you use this package in research, please cite our paper.

@misc{hendi2026geometrypolynomialgroupconvolutional,
  title        = {The Geometry of Polynomial Group Convolutional Neural Networks},
  author       = {Yacoub Hendi and Daniel Persson and Magdalena Larfors},
  year         = {2026},
  eprint       = {2603.29566},
  archivePrefix= {arXiv},
  primaryClass = {cs.LG},
  url          = {https://arxiv.org/abs/2603.29566},
}

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License

MIT License.