Spline representation and redundancies of one-dimensional ReLU neural network models

We analyze the structure of a one-dimensional deep ReLU neural network (ReLU DNN) in comparison to model continuous piecewise linear (CPL) spline functions with arbitrary knots. In particular, we give recursive algorithm transfer parameter set determining DNN into CPL function. Using this representation, show that after removing well-known redundancies DNN, which are caused by positive scaling property, all remaining parameters independent. Moreover, one, two or three hidden layers can represent K arbitrarily prescribed knots (breakpoints), where is number real normalized (up output layer parameters). Our findings useful fix priori conditions on achieve an breakpoints and function values.