Configuration of Continuous Piecewise-Linear Neural Networks

The problem of constructing a general continuous piecewise-linear neural network is considered in this paper. It is shown that every projection domain of an arbitrary continuous piecewise-linear function can be partitioned into convex polyhedra by using difference functions of its local linear functions. Based on these convex polyhedra, a group of continuous piecewise-linear basis functions are formulated. It is proven that a linear combination of these basis functions plus a constant, which we call a standard continuous piecewise-linear neural network, can represent all continuous piecewise-linear functions. In addition, the proposed standard continuous piecewise-linear neural network is applied to solve some function approximation problems. A number of numerical experiments are presented to illustrate that the standard continuous piecewise-linear neural network can be a promising tool for function approximation.