Representing Piecewise-Linear Functions by Functions with Minimal Arity
<|reference_start|>Representing Piecewise-Linear Functions by Functions with Minimal Arity: 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.<|reference_end|>
@article{koutschan2024representing,
    title={Representing Piecewise-Linear Functions by Functions with Minimal Arity},
    author={Christoph Koutschan, Anton Ponomarchuk, Josef Schicho},
    journal={arXiv preprint arXiv:2406.02421},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.02421},
    primaryClass={cs.DM cs.LG cs.SC}
}