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Neural networks are generally built by interleaving (adaptable) linear layerswith (fixed) nonlinear activation functions. To increase their flexibility,several authors have proposed methods for adapting the activation functionsthemselves, endowing them with varying degrees of flexibility. None of theseapproaches, however, have gained wide acceptance in practice, and research inthis topic remains open. In this paper, we introduce a novel family of flexibleactivation functions that are based on an inexpensive kernel expansion at everyneuron. Leveraging over several properties of kernel-based models, we proposemultiple variations for designing and initializing these kernel activationfunctions (KAFs), including a multidimensional scheme allowing to nonlinearlycombine information from different paths in the network. The resulting KAFs canapproximate any mapping defined over a subset of the real line, either convexor nonconvex. Furthermore, they are smooth over their entire domain, linear intheir parameters, and they can be regularized using any known scheme, includingthe use of $\ell_1$ penalties to enforce sparseness. To the best of ourknowledge, no other known model satisfies all these properties simultaneously.In addition, we provide a relatively complete overview on alternativetechniques for adapting the activation functions, which is currently lacking inthe literature. A large set of experiments validates our proposal.

Kafnets: kernel-based non-parametric activation functions for neural networks