Understanding Vector-Valued Neural Networks and Their Relationship with Real and Hypercomplex-Valued Neural Networks

TL;DR

This paper introduces a comprehensive framework for vector-valued neural networks (V-nets), clarifies their relationship with traditional and hypercomplex-valued networks, and demonstrates their implementation in existing deep learning tools.

Contribution

It provides a unifying framework for V-nets, explains their connection to hypercomplex models, and shows practical implementation methods in current deep learning libraries.

Findings

V-nets naturally model intercorrelations between feature channels.

Hypercomplex neural networks are a special case of vector-valued models.

V-nets can be efficiently implemented as real-valued networks in existing frameworks.

Abstract

Despite the many successful applications of deep learning models for multidimensional signal and image processing, most traditional neural networks process data represented by (multidimensional) arrays of real numbers. The intercorrelation between feature channels is usually expected to be learned from the training data, requiring numerous parameters and careful training. In contrast, vector-valued neural networks are conceived to process arrays of vectors and naturally consider the intercorrelation between feature channels. Consequently, they usually have fewer parameters and often undergo more robust training than traditional neural networks. This paper aims to present a broad framework for vector-valued neural networks, referred to as V-nets. In this context, hypercomplex-valued neural networks are regarded as vector-valued models with additional algebraic properties. Furthermore, this paper explains the relationship between vector-valued and traditional neural networks. Precisely, a vector-valued neural network can be obtained by placing restrictions on a real-valued model to consider the intercorrelation between feature channels. Finally, we show how V-nets, including hypercomplex-valued neural networks, can be implemented in current deep-learning libraries as real-valued networks.