@article {533,
	title = {Kafnets: Kernel-based non-parametric activation functions for neural networks},
	journal = {Neural Networks},
	volume = {110},
	year = {2019},
	month = {February},
	pages = {19-32},
	doi = {10.1016/j.neunet.2018.11.002},
	author = {Scardapane, Simone and Van Vaerenbergh, Steven and Totaro, Simone and Uncini, Aurelio}
}
@article {530,
	title = {Complex-valued Neural Networks with Non-parametric Activation Functions},
	journal = {IEEE Transactions on Emerging Topics in Computational Intelligence},
	volume = {3},
	year = {2018},
	abstract = {Complex-valued neural networks (CVNNs) are a powerful modeling tool for domains where data can be naturally interpreted in terms of complex numbers. However, several analytical properties of the complex domain (such as holomorphicity) make the design of CVNNs a more challenging task than their real counterpart. In this paper, we consider the problem of flexible activation functions (AFs) in the complex domain, i.e., AFs endowed with sufficient degrees of freedom to adapt their shape given the training data. While this problem has received considerable attention in the real case, very limited literature exists for CVNNs, where most activation functions are generally developed in a split fashion (i.e., by considering the real and imaginary parts of the activation separately) or with simple phase-amplitude techniques. Leveraging over the recently proposed kernel activation functions, and related advances in the design of complex-valued kernels, we propose the first fully complex, nonparametric activation function for CVNNs, which is based on a kernel expansion with a fixed dictionary that can be implemented efficiently on vectorized hardware. Several experiments on common use cases, including prediction and channel equalization, validate our proposal when compared to real-valued neural networks and CVNNs with fixed activation functions.},
	issn = {2471-285X},
	doi = {10.1109/TETCI.2018.2872600},
	url = {https://ieeexplore.ieee.org/document/8495012},
	author = {Scardapane, Simone and Van Vaerenbergh, Steven and Hussain, Amir and Uncini, Aurelio}
}
@conference {525,
	title = {Recurrent Neural Networks With Flexible Gates Using Kernel Activation Functions},
	booktitle = {IEEE International Workshop on Machine Learning for Signal Processing},
	year = {2018},
	month = {September},
	publisher = {IEEE},
	organization = {IEEE},
	address = {Aalborg, Denmark},
	abstract = {Gated recurrent neural networks have achieved remarkable results in the analysis of sequential data. Inside these networks, gates are used to control the flow of information, allowing to model even very long-term dependencies in the data. In this paper, we investigate whether the original gate equation (a linear projection followed by an element-wise sigmoid) can be improved. In particular, we design a more flexible architecture, with a small number of adaptable parameters, which is able to model a wider range of gating functions than the classical one. To this end, we replace the sigmoid function in the standard gate with a non-parametric formulation extending the recently proposed kernel activation function (KAF), with the addition of a residual skip-connection. A set of experiments on sequential variants of the MNIST dataset shows that the adoption of this novel gate allows to improve accuracy with a negligible cost in terms of computational power and with a large speed-up in the number of training iterations.},
	author = {Scardapane, Simone and Van Vaerenbergh, Steven and Comminiello, Danilo and Totaro, Simone and Uncini, Aurelio}
}
@conference {521,
	title = {Improving Graph Convolutional Networks with Non-Parametric Activation Functions},
	booktitle = {26th European Signal Processing Conference (EUSIPCO 2018)},
	year = {2018},
	month = {September},
	publisher = {EURASIP},
	organization = {EURASIP},
	address = {Rome, Italy},
	abstract = {Graph neural networks (GNNs) are a class of neural networks that allow to efficiently perform inference on data that is associated to a graph structure, such as, e.g., citation networks or knowledge graphs. While several variants of GNNs have been proposed, they only consider simple nonlinear activation functions in their layers, such as rectifiers or squashing functions. In this paper, we investigate the use of graph convolutional networks (GCNs) when combined with more complex activation functions, able to adapt from the training data. More specifically, we extend the recently proposed kernel activation function, a non-parametric model which can be implemented easily, can be regularized with standard l?p-norms techniques, and is smooth over its entire domain. Our experimental evaluation shows that the proposed architecture can significantly improve over its baseline, while similar improvements cannot be obtained by simply increasing the depth or size of the original GCN.},
	author = {Scardapane, Simone and Van Vaerenbergh, Steven and Comminiello, Danilo and Uncini, Aurelio}
}
