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.

翻译：任何连续分段线性函数 $F\colon \mathbb{R}^{n}\to \mathbb{R}$ 均可表示为至多 $n+1$ 个仿射线性函数的 $\max$ 函数的线性组合。在我们先前的论文[“用低元数函数表示分段线性函数”，AAECC, 2023]中，我们证明了 $n+1$ 个参数的这个上界是紧的。在本文中，我们通过建立函数 $F$ 与其任意此类分解所需的最小参数数量之间的对应关系，扩展了这一结果。我们证明了由函数 $F$ 诱导的输入空间 $\mathbb{R}^{n}$ 的剖分与 $\max$ 函数中的参数数量存在直接联系。