Representing Piecewise-Linear Functions by Functions with Minimal Arity

TL;DR

This paper explores how continuous piecewise-linear functions can be efficiently represented using minimal-arity max functions, linking the function's space tessellation to the minimal number of arguments needed.

Contribution

It extends previous work by establishing a correspondence between the function's tessellation and the minimal number of arguments in max function decompositions.

Findings

The minimal number of arguments in max functions corresponds to the tessellation complexity of the function.

The paper proves the tightness of the upper bound of n+1 arguments for representing such functions.

A direct connection between input space tessellation and max function decomposition is established.

Abstract

Any continuous piecewise-linear function $F\colon \mathbb{R}^{n}\to \mathbb{R}$ can be represented as a linear combination of $\max$ functions of at most $n+1$ affine-linear functions. In our previous paper [``Representing piecewise linear functions by functions with small arity'', AAECC, 2023], we showed that this upper bound of $n+1$ arguments is tight. In the present paper, we extend this result by establishing a correspondence between the function $F$ and the minimal number of arguments that are needed in any such decomposition. We show that the tessellation of the input space $\mathbb{R}^{n}$ induced by the function $F$ has a direct connection to the number of arguments in the $\max$ functions.