Triangulating PL functions and the existence of efficient ReLU DNNs
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910937973260288 |
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| author | Calegari, Danny |
| author_facet | Calegari, Danny |
| contents | We show that every piecewise linear function $f:R^d \to R$ with compact support a polyhedron $P$ has a representation as a sum of so-called `simplex functions'. Such representations arise from degree 1 triangulations of the relative homology class (in $R^{d+1}$) bounded by $P$ and the graph of $f$, and give a short elementary proof of the existence of efficient universal ReLU neural networks that simultaneously compute all such functions $f$ of bounded complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07137 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Triangulating PL functions and the existence of efficient ReLU DNNs Calegari, Danny Machine Learning Geometric Topology We show that every piecewise linear function $f:R^d \to R$ with compact support a polyhedron $P$ has a representation as a sum of so-called `simplex functions'. Such representations arise from degree 1 triangulations of the relative homology class (in $R^{d+1}$) bounded by $P$ and the graph of $f$, and give a short elementary proof of the existence of efficient universal ReLU neural networks that simultaneously compute all such functions $f$ of bounded complexity. |
| title | Triangulating PL functions and the existence of efficient ReLU DNNs |
| topic | Machine Learning Geometric Topology |
| url | https://arxiv.org/abs/2505.07137 |