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.

翻译：尽管深度学习模型在多维信号和图像处理领域取得了许多成功应用，但大多数传统神经网络处理的是由（多维）实数数组表示的数据。特征通道间的相互关联通常期望从训练数据中学习得到，这就需要大量参数和精细的训练过程。相比之下，向量值神经网络旨在处理向量数组，并能自然地考虑特征通道间的相互关联。因此，这类网络通常参数更少，且训练过程往往比传统神经网络更稳健。本文旨在提出一个统一的向量值神经网络框架，称之为V-nets。在此框架下，超复值神经网络被视为具有额外代数性质的向量值模型。此外，本文阐明了向量值神经网络与传统神经网络之间的关系。具体而言，向量值神经网络可通过在实值模型上施加约束来考虑特征通道间的相互关联而得到。最后，我们展示了如何将V-nets（包括超复值神经网络）作为实值网络在当前深度学习库中实现。