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Is it possible to train a simple perceptron with a threshold activation function such as this one: https://en.wikipedia.org/wiki/Perceptron with Backpropagation instead of the perceptron rule?

is it a problem that threshold functions are not defined around 0?

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No, it is not possible. In order to use backprop you have to have non-zero derivatives of the function. Derivative of the step function is zero everywhere.

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