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Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators

Created by
  • Haebom

Author

Sven Dummer, Tjeerd Jan Heeringa, Jos e A. Iglesias

Outline

This paper focuses on $\mathbb{R}^d$-valued neural networks and neural operator models, particularly on $\mathbb{R}^d$-valued neural networks. To address the limitations of previous research on reproducible kernel Banach spaces (RKBSs), we develop a general definition of vector-valued RKBS (vv-RKBS) without constraints such as symmetric kernel domains, finite-dimensional output spaces, reflexivity, or divisibility. We show that shallow $\mathbb{R}^d$-valued neural networks belong to a specific vv-RKBS, and we also prove that neural operator models such as DeepONet and Hypernetwork architectures belong to integral and neural vv-RKBSs. Finally, we present the Representative Theorem, which states that optimizations in this function space recover the corresponding neural architectures.

Takeaways, Limitations

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Takeaways:
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We developed a general definition of vector-valued reproducible kernel Banach space (vv-RKBS) to relax the constraints of existing definitions.
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We prove that neural networks and neural operator models (DeepONet, Hypernetwork) with shallow $\mathbb{R}^d$-values belong to a specific vv-RKBS.
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We show that optimization in function space recovers neural architectures through the Representer Theorem.
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Limitations:
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There may be a lack of in-depth analysis of specific experimental results or real-world application cases.
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Further research is needed to determine how broadly the presented theoretical results can be applied to various neural network architectures.
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Analysis of other types of neural network architectures not covered in this paper is needed.
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