  The paper proposes representation functionals in a dual paradigm where
learning jointly concerns both linear convolutional weights and parametric
forms of nonlinear activation functions. The nonlinear forms proposed for
performing the functional representation are associated with a new class of
parametric neural transfer functions called rectified power sigmoid units. This
class is constructed to integrate both advantages of sigmoid and rectified
linear unit functions, in addition with rejecting the drawbacks of these
functions. Moreover, the analytic form of this new neural class involves scale,
shift and shape parameters so as to obtain a wide range of activation shapes,
including the standard rectified linear unit as a limit case. Parameters of
this neural transfer class are considered as learnable for the sake of
discovering the complex shapes that can contribute in solving machine learning
issues. Performance achieved by the joint learning of convolutional and
rectified power sigmoid learnable parameters are shown outstanding in both
shallow and deep learning frameworks. This class opens new prospects with
respect to machine learning in the sense that learnable parameters are not only
attached to linear transformations, but also to suitable nonlinear operators.
