PHNNs: Lightweight Neural Networks via Parameterized Hypercomplex Convolutions

TL;DR

This paper introduces parameterized hypercomplex neural networks (PHNNs), which are lightweight, flexible, and efficient models that adapt hypercomplex convolutional layers directly from data, outperforming real and quaternion-based models across multiple domains.

Contribution

The paper proposes a novel parameterization of hypercomplex convolutional layers, creating a flexible family of PHNNs that operate efficiently in any domain without predefined algebraic structures.

Findings

PHNNs operate with 1/n parameters compared to real domain models.

PHNNs outperform real and quaternion neural networks on image and audio datasets.

The method adapts hypercomplex convolutions directly from data, enhancing flexibility and efficiency.

Abstract

Hypercomplex neural networks have proven to reduce the overall number of parameters while ensuring valuable performance by leveraging the properties of Clifford algebras. Recently, hypercomplex linear layers have been further improved by involving efficient parameterized Kronecker products. In this paper, we define the parameterization of hypercomplex convolutional layers and introduce the family of parameterized hypercomplex neural networks (PHNNs) that are lightweight and efficient large-scale models. Our method grasps the convolution rules and the filter organization directly from data without requiring a rigidly predefined domain structure to follow. PHNNs are flexible to operate in any user-defined or tuned domain, from 1D to $n$D regardless of whether the algebra rules are preset. Such a malleability allows processing multidimensional inputs in their natural domain without annexing further dimensions, as done, instead, in quaternion neural networks for 3D inputs like color images. As a result, the proposed family of PHNNs operates with $1/n$ free parameters as regards its analog in the real domain. We demonstrate the versatility of this approach to multiple domains of application by performing experiments on various image datasets as well as audio datasets in which our method outperforms real and quaternion-valued counterparts. Full code is available at: https://github.com/eleGAN23/HyperNets.