Understanding Vector-Valued Neural Networks and Their Relationship with
  Real and Hypercomplex-Valued Neural Networks
Marcos Eduardo Valle
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.