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Neural network activation functions

There is a whole cottage industry in showing neural networks are reasonably universal function approximators with fairly general nonlinearities as activations, under fairly general conditions. Nonetheless, you might like to play with the precise form of the nonlinearities, even making them themselves directly learnable, because some function shapes might have better approximation properties in a sense I will not trouble to make rigorous now, vague hand-waving arguments being the whole point of deep learning.

Nonetheless, here are a some handy references.