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Representing Piecewise-Linear Functions by Functions with Minimal Arity
[Submitted on 4 Jun 2024 (v1), last revised 17 Jun 2026 (this ve · 2026-06-19 · via cs.LG updates on arXiv.org

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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.

Submission history

From: Anton Ponomarchuk [view email]
[v1] Tue, 4 Jun 2024 15:39:08 UTC (7,451 KB)
[v2] Wed, 17 Jun 2026 21:50:44 UTC (1,116 KB)

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