  Complex-valued neural networks (CVNNs) are an emerging field of research in
neural networks due to their potential representational properties for audio,
image, and physiological signals. It is common in signal processing to
transform sequences of real values to the complex domain via a set of complex
basis functions, such as the Fourier transform. We show how CVNNs can be used
to learn complex representations of real valued time-series data. We present
methods and results using a framework that can compose holomorphic and
non-holomorphic functions in a multi-layer network using a theoretical result
called the Wirtinger derivative. We test our methods on a representation
learning task for real-valued signals, recurrent complex-valued networks and
their real-valued counterparts. Our results show that recurrent complex-valued
networks can perform as well as their real-valued counterparts while learning
filters that are representative of the domain of the data.
