Representing Piecewise-Linear Functions by Functions with Minimal Arity
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911903638355968 |
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| author | Koutschan, Christoph Ponomarchuk, Anton Schicho, Josef |
| author_facet | Koutschan, Christoph Ponomarchuk, Anton Schicho, Josef |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02421 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representing Piecewise-Linear Functions by Functions with Minimal Arity Koutschan, Christoph Ponomarchuk, Anton Schicho, Josef Discrete Mathematics Machine Learning Symbolic Computation 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. |
| title | Representing Piecewise-Linear Functions by Functions with Minimal Arity |
| topic | Discrete Mathematics Machine Learning Symbolic Computation |
| url | https://arxiv.org/abs/2406.02421 |