  Complex-valued neural networks (CVNNs) have been shown to be powerful
nonlinear approximators when the input data can be properly modeled in the
complex domain. One of the major challenges in scaling up CVNNs in practice is
the design of complex activation functions. Recently, we proposed a novel
framework for learning these activation functions neuron-wise in a
data-dependent fashion, based on a cheap one-dimensional kernel expansion and
the idea of kernel activation functions (KAFs). In this paper we argue that,
despite its flexibility, this framework is still limited in the class of
functions that can be modeled in the complex domain. We leverage the idea of
widely linear complex kernels to extend the formulation, allowing for a richer
expressiveness without an increase in the number of adaptable parameters. We
test the resulting model on a set of complex-valued image classification
benchmarks. Experimental results show that the resulting CVNNs can achieve
higher accuracy while at the same time converging faster.
