We present a novel semi-supervised two-step training procedure for our discontinuous deep learning model, tailored to its structure, and we provide theoretical support for its effectiveness. A piecewise function is said to be continuous when there are an infinite number of points on the graph that are always connecting one function to the next. Firstly a Piecewise Continuous function is made up of a finite number of continuous pieces on each finite subinterval 0, T. We provide approximation guarantees for our architecture in the space of bounded continuous functions and universal approximation guarantees in the space of piecewise continuous functions which we introduced herein.

We propose a new discontinuous deep neural network model trainable via a decoupled two-step procedure that avoids passing gradient updates through the network’s only and strategically placed, discontinuous unit. Buy 4 avocadoes for 1.98 a piece, and get each one after that for half off. Just the other day, you were at Lunds, where they were trying to sell some extra avocadoes, so they had a promotion going on. This paper focuses on the case where the discontinuities arise from distinct sub-patterns, each defined on different parts of the input space. It is a problem with types, you have to change the following line, so that the x is given as floats: x np.array ( -3, -2, -1, 0, 1, 2, 3).astype (np.float) otherwise the piecewiselinear will might end up casting the types. Continuity and Piecewise Functions Name: Now that you’re in calculus class, you’re always seeing the math wherever you go. Most stochastic gradient descent algorithms can optimize neural networks that are sub-differentiable in their parameters however, this implies that the neural network’s activation function must exhibit a degree of continuity which limits the neural network model’s uniform approximation capacity to continuous functions.

Continuous piecewise function