Report
On best approximation by multivariate ridge functions with applications to generalized translation networks
| العنوان: | On best approximation by multivariate ridge functions with applications to generalized translation networks |
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| المؤلفون: | Geuchen, Paul, Salanevich, Palina, Schavemaker, Olov, Voigtlaender, Felix |
| سنة النشر: | 2024 |
| المجموعة: | Computer Science Mathematics Statistics |
| مصطلحات موضوعية: | Mathematics - Functional Analysis, Computer Science - Machine Learning, Statistics - Machine Learning, 41A30, 41A25, 41A63, 46E35, 68T07 |
| الوصف: | We prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., functions of the form $\mathbb{R}^d \ni x \mapsto \sum_{k=1}^n h_k(A_k x) \in \mathbb{R}$ with $h_k : \mathbb{R}^\ell \to \mathbb{R}$ and $A_k \in \mathbb{R}^{\ell \times d}$. We show that the order of approximation asymptotically behaves as $n^{-r/(d-\ell)}$, where $r$ is the regularity of the Sobolev functions to be approximated. Our lower bound even holds when approximating $L^\infty$-Sobolev functions of regularity $r$ with error measured in $L^1$, while our upper bound applies to the approximation of $L^p$-Sobolev functions in $L^p$ for any $1 \leq p \leq \infty$. These bounds generalize well-known results about the approximation properties of univariate ridge functions to the multivariate case. Moreover, we use these bounds to obtain sharp asymptotic bounds for the approximation of Sobolev functions using generalized translation networks and complex-valued neural networks. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2412.08453 |
| رقم الانضمام: | edsarx.2412.08453 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |