  We propose a new structure for the complex-valued autoencoder by introducing
additional degrees of freedom into its design through a widely linear (WL)
transform. The corresponding widely linear backpropagation algorithm is also
developed using the $\mathbb{CR}$ calculus, to unify the gradient calculation
of the cost function and the underlying WL model. More specifically, all the
existing complex-valued autoencoders employ the strictly linear transform,
which is optimal only when the complex-valued outputs of each network layer are
independent of the conjugate of the inputs. In addition, the widely linear
model which underpins our work allows us to consider all the second-order
statistics of inputs. This provides more freedom in the design and enhanced
optimization opportunities, as compared to the state-of-the-art. Furthermore,
we show that the most widely adopted cost function, i.e., the mean squared
error, is not best suited for the complex domain, as it is a real quantity with
a single degree of freedom, while both the phase and the amplitude information
need to be optimized. To resolve this issue, we design a new cost function,
which is capable of controlling the balance between the phase and the amplitude
contribution to the solution. The experimental results verify the superior
performance of the proposed autoencoder together with the new cost function,
especially for the imaging scenarios where the phase preserves extensive
information on edges and shapes.
