hypercomplex-neural-network

Overview

This repository contains the implementation and experimental results for a comparative study of quaternion neural networks (QNNs) and their generalizations to Clifford/Parameterized Hypercomplex Neural Networks (PHNNs), including PHB-cos models. In this work we moved beyond quaternions and observed tangible benefits in accuracy, efficiency, and interpretability.

Contents

• report/ – LaTeX source and compiled PDF of the report.
• code/ – Implementation of real, quaternion, PHC, and PHB-cos models, training scripts, and evaluation routines.
• plots/ – Generated figures (accuracy vs parameters, validation accuracy, explanation IoU).
• results/ – Aggregated CSVs and LaTeX tables from 5-seed experiments.

Tutorials

The folder tutorials/ contains a set of tutorials to understand the Parameterized Hypercomplex Multiplication (PHM) layer and the Parameterized Hypercomplex Convolutional (PHC) layer. We develop simple toy examples to learn the matrices A that define algebra rules in order to demonstrate the effectiveness of the proposed approach.

• PHM tutorial.ipynb is a simple tutorial which shows how the PHM layer learns the Hamilton product between two pure quaternions.
• PHC tutorial.ipynb is a simple tutorial which shows how the PHC layer learn the Hamilton rule to organize filters in convolution.
• Toy regression examples with PHM.ipynb is a notebook containing some regression tasks.

Key Results

• PHC models (especially $n=6$) achieve higher accuracy with fewer parameters compared to quaternion and real-valued networks.
• PHB-cos models provide state-of-the-art interpretability by aligning filters with inputs, yielding better IoU with ground-truth ROIs compared to Grad-CAM/LIME explanations.
• Parameter efficiency scales with $1/n$, validating the advantage of Clifford-based approaches over quaternions.

Installation

git clone https://github.com/Astraflaneur/hypercomplex-comparison.git
cd hypercomplex-neural-network
pip install -r ./code/requirements.txt

Usage

To train a model (example: PHCNet with $n=6$ on CIFAR-10):

python train.py --model phc --n 6 --dataset cifar10 --epochs 200 --seeds 5

To reproduce plots and tables:

python make_plots_published.py --csv results.csv --outdir outputs