Erzeugunsgrad, VC-Dimension and Neural Networks with rational activation function
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915244081676288 |
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| author | Pardo, Luis Miguel Sebastián, Daniel |
| author_facet | Pardo, Luis Miguel Sebastián, Daniel |
| contents | The notion of Erzeugungsgrad was introduced by Joos Heintz in 1983 to bound the number of non-empty cells occurring after a process of quantifier elimination. We extend this notion and the combinatorial bounds of Theorem 2 in Heintz (1983) using the degree for constructible sets defined in Pardo-Sebastián (2022). We show that the Erzeugungsgrad is the key ingredient to connect affine Intersection Theory over algebraically closed fields and the VC-Theory of Computational Learning Theory for families of classifiers given by parameterized families of constructible sets. In particular, we prove that the VC-dimension and the Krull dimension are linearly related up to logarithmic factors based on Intersection Theory. Using this relation, we study the density of correct test sequences in evasive varieties. We apply these ideas to analyze parameterized families of neural networks with rational activation function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Erzeugunsgrad, VC-Dimension and Neural Networks with rational activation function Pardo, Luis Miguel Sebastián, Daniel Machine Learning Algebraic Geometry 13F20, 14A10, 68T07 The notion of Erzeugungsgrad was introduced by Joos Heintz in 1983 to bound the number of non-empty cells occurring after a process of quantifier elimination. We extend this notion and the combinatorial bounds of Theorem 2 in Heintz (1983) using the degree for constructible sets defined in Pardo-Sebastián (2022). We show that the Erzeugungsgrad is the key ingredient to connect affine Intersection Theory over algebraically closed fields and the VC-Theory of Computational Learning Theory for families of classifiers given by parameterized families of constructible sets. In particular, we prove that the VC-dimension and the Krull dimension are linearly related up to logarithmic factors based on Intersection Theory. Using this relation, we study the density of correct test sequences in evasive varieties. We apply these ideas to analyze parameterized families of neural networks with rational activation function. |
| title | Erzeugunsgrad, VC-Dimension and Neural Networks with rational activation function |
| topic | Machine Learning Algebraic Geometry 13F20, 14A10, 68T07 |
| url | https://arxiv.org/abs/2504.11345 |