Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909817732333568 |
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| author | Dummer, Sven Heeringa, Tjeerd Jan Iglesias, José A. |
| author_facet | Dummer, Sven Heeringa, Tjeerd Jan Iglesias, José A. |
| contents | Recently, there has been growing interest in characterizing the function spaces underlying neural networks. While shallow and deep scalar-valued neural networks have been linked to scalar-valued reproducing kernel Banach spaces (RKBS), $\mathbb{R}^d$-valued neural networks and neural operator models remain less understood in the RKBS setting. To address this gap, we develop a general definition of vector-valued RKBS (vv-RKBS), which inherently includes the associated reproducing kernel. Our construction extends existing definitions by avoiding restrictive assumptions such as symmetric kernel domains, finite-dimensional output spaces, reflexivity, or separability, while still recovering familiar properties of vector-valued reproducing kernel Hilbert spaces (vv-RKHS). We then show that shallow $\mathbb{R}^d$-valued neural networks are elements of a specific vv-RKBS, namely an instance of the integral and neural vv-RKBS. To also explore the functional structure of neural operators, we analyze the DeepONet and Hypernetwork architectures and demonstrate that they too belong to an integral and neural vv-RKBS. In all cases, we establish a Representer Theorem, showing that optimization over these function spaces recovers the corresponding neural architectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators Dummer, Sven Heeringa, Tjeerd Jan Iglesias, José A. Functional Analysis Artificial Intelligence Machine Learning 46E15, 68T07, 46G10, 46E22, 46B10, 26B40 G.1.2; G.1.6; I.5.1; I.2.6 Recently, there has been growing interest in characterizing the function spaces underlying neural networks. While shallow and deep scalar-valued neural networks have been linked to scalar-valued reproducing kernel Banach spaces (RKBS), $\mathbb{R}^d$-valued neural networks and neural operator models remain less understood in the RKBS setting. To address this gap, we develop a general definition of vector-valued RKBS (vv-RKBS), which inherently includes the associated reproducing kernel. Our construction extends existing definitions by avoiding restrictive assumptions such as symmetric kernel domains, finite-dimensional output spaces, reflexivity, or separability, while still recovering familiar properties of vector-valued reproducing kernel Hilbert spaces (vv-RKHS). We then show that shallow $\mathbb{R}^d$-valued neural networks are elements of a specific vv-RKBS, namely an instance of the integral and neural vv-RKBS. To also explore the functional structure of neural operators, we analyze the DeepONet and Hypernetwork architectures and demonstrate that they too belong to an integral and neural vv-RKBS. In all cases, we establish a Representer Theorem, showing that optimization over these function spaces recovers the corresponding neural architectures. |
| title | Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators |
| topic | Functional Analysis Artificial Intelligence Machine Learning 46E15, 68T07, 46G10, 46E22, 46B10, 26B40 G.1.2; G.1.6; I.5.1; I.2.6 |
| url | https://arxiv.org/abs/2509.26371 |