  Generative adversarial nets (GANs) have become a preferred tool for tasks
involving complicated distributions. To stabilise the training and reduce the
mode collapse of GANs, one of their main variants employs the integral
probability metric (IPM) as the loss function. This provides extensive IPM-GANs
with theoretical support for basically comparing moments in an embedded domain
of the \textit{critic}. We generalise this by comparing the distributions
rather than their moments via a powerful tool, i.e., the characteristic
function (CF), which uniquely and universally comprising all the information
about a distribution. For rigour, we first establish the physical meaning of
the phase and amplitude in CF, and show that this provides a feasible way of
balancing the accuracy and diversity of generation. We then develop an
efficient sampling strategy to calculate the CFs. Within this framework, we
further prove an equivalence between the embedded and data domains when a
reciprocal exists, where we naturally develop the GAN in an auto-encoder
structure, in a way of comparing everything in the embedded space (a
semantically meaningful manifold). This efficient structure uses only two
modules, together with a simple training strategy, to achieve bi-directionally
generating clear images, which is referred to as the reciprocal CF GAN
(RCF-GAN). Experimental results demonstrate the superior performances of the
proposed RCF-GAN in terms of both generation and reconstruction.
