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Erzeugunsgrad, VC-Dimension and Neural Networks with rational activation function
Machine Learning · Submitted 15 Apr 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In particular, we prove that the VC-dimension and the Krull dimension are linearly related up to logarithmic factors based on Intersection Theory.
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The abstract, sentence by sentence
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-Sebasti\'an (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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | We extend this notion and the combinatorial bounds of Theorem 2 in Heintz (1983) using the degree for constructible sets defined in Pardo-Sebasti\'an (2022). | contribution 0.95 | 0.92 | 0.07 | asserted 1.00 |
| s2 | 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. | result 0.48 | 0.91 | 0.59 | asserted 1.00 |
| s3 | In particular, we prove that the VC-dimension and the Krull dimension are linearly related up to logarithmic factors based on Intersection Theory. | result 0.99 | 0.92 | 0.38 | asserted 1.00 |
| s4 | Using this relation, we study the density of correct test sequences in evasive varieties. | contribution 0.92 | 0.84 | 0.06 | asserted 0.88 |
| s5 | We apply these ideas to analyze parameterized families of neural networks with rational activation function. | contribution 0.95 | 0.88 | 0.11 | asserted 0.98 |
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Original abstract
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-Sebasti\'an (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.
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Cites as background 2 papers
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A tropical algebraic rational function is the difference between two tropical polynomials (cf. [ref] and references therein).
Decomposing tropical rational functions · Examples of activation functions: ReLUs, evaluation schemes, polynomials, rationals, Nash functions and Pfaffian functio · 0.93
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The class $$ of activation functions is formed by a single function, sometimes denoted as $(·)_+$ (cf. [ref] and references therein): [equation] The class $_∘^(n)[]$ of potentially evaluable functions by a ReLU neural network with $n$ input nodes ${X_1,…, X_n}$ and a single output node is an extension of the class of tropical algebraic rational functions (cf. [ref] , [ref] , [ref] , [ref] or [ref] and references therein for further information) admitting real coefficients.
Tropical Newton-Puiseux polynomials · Examples of activation functions: ReLUs, evaluation schemes, polynomials, rationals, Nash functions and Pfaffian functio · 0.89
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