Title: Learning Neuron Non-Linearities with Kernel-Based Deep Neural Networks. Abstract: The effectiveness of deep neural architectures has been widely supported in terms of both experimental and foundational principles. There is also clear evidence that the activation function (e.g. the rectiﬁer and the LSTM units) plays a crucial role in the complexity of learning. Based on this remark, this paper discusses an optimal selection of the neuron non-linearity in a functional framework that is inspired from classic regularization arguments. A representation theorem is given which indicates that the best activation function is a kernel expansion in the training set, that can be effectively approximated over an opportune set of points modeling 1-D clusters. The idea can be naturally extended to recurrent networks, where the expressiveness of kernel-based activation functions turns out to be a crucial ingredient to capture long-term dependencies. We give experimental evidence of this property by a set of challenging experiments, where we compare the results with neural architectures based on state of the art LSTM cells.