Trainable Compound Activation Functions for Machine Learning

Activation functions (AF) are necessary components of neural networks that allow approximation of functions, but AFs in current use are usually simple monotonically increasing functions. In this paper, we propose trainable compound AF (TCA) composed of a sum of shifted and scaled simple AFs. TCAs increase the effectiveness of networks with fewer parameters compared to added layers. TCAs have a special interpretation in generative networks because they effectively estimate the marginal distributions of each dimension of the data using a mixture distribution, reducing modality and making linear dimension reduction more effective. When used in restricted Boltzmann machines (RBMs), they result in a novel type of RBM with mixture-based stochastic units. Improved performance is demonstrated in experiments using RBMs, deep belief networks (DBN), projected belief networks (PBN), and variational auto-encoders (VAE).