Complex-valued neural networks (CVNNs) have recently been successful in
various pioneering areas which involve wave-typed information and
frequency-domain processing. This work addresses different structures and
classification of CVNNs. The theory behind complex activation functions,
implications related to complex differentiability and special activations for
CVNN output layers are presented. The work also discusses CVNN learning and
optimization using gradient and non-gradient based algorithms. Complex
Backpropagation utilizing complex chain rule is also explained in terms of
Wirtinger calculus. Moreover, special modules for building CVNN models, such as
complex batch normalization and complex random initialization are also
discussed. The work also highlights libraries and software blocks proposed for
CVNN implementations and discusses future directions. The objective of this
work is to understand the dynamics and most recent developments of CVNNs.

复杂值神经网络（CVNNs）涵盖了波状信息和频域处理的不同结构和分类，并解释了复杂激活函数、复杂可微性的相关含义以及 CVNN 输出层的特殊激活。还讨论了使用基于梯度和非梯度的算法进行 CVNN 学习和优化的情况。此外，通过 Wirtinger 微积分术语解释了利用复杂链规则的复杂反向传播。还讨论了用于构建 CVNN 模型的特殊模块，如复杂批归一化和复杂随机初始化。该工作还强调了提出的用于 CVNN 实现的库和软件模块，并讨论了未来的发展方向。本研究的目标是理解 CVNN 的动态和最新发展。

复数神经网络 ---- 理论与分析

Complex-valued Neural Networks -- Theory and Analysis

Artificial neural networks (ANNs) based machine learning models and
especially deep learning models have been widely applied in computer vision,
signal processing, wireless communications, and many other domains, where
complex numbers occur either naturally or by design. However, most of the
current implementations of ANNs and machine learning frameworks are using real
numbers rather than complex numbers. There are growing interests in building
ANNs using complex numbers, and exploring the potential advantages of the
so-called complex-valued neural networks (CVNNs) over their real-valued
counterparts. In this paper, we discuss the recent development of CVNNs by
performing a survey of the works on CVNNs in the literature. Specifically, a
detailed review of various CVNNs in terms of activation function, learning and
optimization, input and output representations, and their applications in tasks
such as signal processing and computer vision are provided, followed by a
discussion on some pertinent challenges and future research directions.

本文综述了近期在文献中对 CVNNs 的各种工作，并详细研究了各种 CVNNs 的激活功能、学习与优化、输入和输出表示，以及在信号处理和计算机视觉等任务中的应用，同时探讨了一些挑战和未来研究方向。