Unification of popular artificial neural network activation functions

We present a unified representation of the most popular neural network activation functions. Adopting Mittag-Leffler functions of fractional calculus, we propose a flexible and compact functional form that is able to interpolate between various activation functions and mitigate common problems in training neural networks such as vanishing and exploding gradients. The presented gated representation extends the scope of fixed-shape activation functions to their adaptive counterparts whose shape can be learnt from the training data. The derivatives of the proposed functional form can also be expressed in terms of Mittag-Leffler functions making it a suitable candidate for gradient-based backpropagation algorithms. By training LeNet-5 neural network on MNIST and CIFAR-10 datasets, we demonstrate that adopting a unified gated representation of activation functions offers a promising and affordable alternative to individual built-in implementations of activation functions in conventional machine learning frameworks.