Representing piecewise linear functions by functions with small arity

Abstract

A piecewise linear function can be described in different forms: as a nested expression of \(\min\)- and \(\max\)-functions, as a difference of two convex piecewise linear functions, or as a linear combination of maxima of affine-linear functions. In this paper, we provide two main results: first, we show that for every piecewise linear function \(f:{\mathbb{R}}^{n} \rightarrow {\mathbb{R}}\), there exists a linear combination of \(\max\)-functions with at most \(n+1\) arguments, and give an algorithm for its computation. Moreover, these arguments are contained in the finite set of affine-linear functions that coincide with the given function in some open set. Second, we prove that the piecewise linear function \(\max (0, x_{1}, \ldots , x_{n})\) cannot be represented as a linear combination of maxima of less than \(n+1\) affine-linear arguments. This was conjectured by Wang and Sun (IEEE Trans Inf Theory 51:4425–4431, 2005) in a paper on representations of piecewise linear functions as linear combination of maxima.