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arXiv · 2406.02421

Representing Piecewise-Linear Functions by Functions with Minimal Arity

Christoph Koutschan·Anton Ponomarchuk·Josef Schicho
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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.

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Christoph Koutschan, Anton Ponomarchuk, Josef Schicho. 2024-06-04. Representing Piecewise-Linear Functions by Functions with Minimal Arity. https://arxiv.org/abs/2406.02421

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