From Brainwave to Breakthrough: Exploring Quaternion and Octonion Neural Networks in Deep Learning

Introduction

From Inspiration to Implementation

As I dove deeper into the world of neural networks, an idea began to form — I had a brainwave. Having studied neural networks for some time, I became increasingly fascinated by the ways in which higher-dimensional number systems, such as quaternions and octonions, had been applied to fields like 3D computer graphics and quantum mechanics. Quaternions, which are a 4-dimensional extension of complex numbers, have long been used to represent rotations in 3D space, while octonions take this even further, expanding into 8 dimensions and being explored in theoretical physics.

While reading through the mathematical literature on these hypercomplex numbers, I started to wonder: could these powerful algebraic systems be applied to neural networks in a way that enhances their ability to process complex, high-dimensional data? What if the efficiencies inherent in quaternion and octonion arithmetic could reduce the complexity of modern neural networks while maintaining (or even improving) their performance?

The Application

This led me to experiment with incorporating quaternion and octonion layers into…