Representing Piecewise-Linear Functions by Functions With Minimal Arity
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.
Fields of Study & Subjects
Advanced Optimization Algorithms Research
Publication details
- Journal / source
- IEEE Transactions on Information Theory
- Volume
- 72
- Issue
- 9
- Pages
- 6825–6831
- Publication date
- 19 June 2026
- Language
- EN