  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.
