aiqbench analysis
This research establishes a rigorous mathematical framework by defining adjoint pairs of vector-valued reproducing kernel Banach spaces to characterize the function spaces underlying $\mathbb{R}^d$-valued neural networks and operators like DeepONet. By proving that these architectures belong to integral vv-RKBSs without restrictive assumptions, the work provides a representer theorem essential for understanding optimization over complex vector-valued output structures in machine learning theory.
Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators
arXiv:2509.26371v3 Announce Type: replace-cross Abstract: 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 notion of adjoint pairs of vector-valued RKBSs (vv-RKBS), which inherently involves an associated reproducing kernel, and prove that every vv-RKBS belongs to such a pair. Our construction extends existing kernel 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 an integral 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 vv-RKBS. In all cases, we establish a representer theorem, showing that optimization over these function spaces recovers the corresponding neural architectures.
Abstract: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 notion of adjoint pairs of vector-valued RKBSs (vv-RKBS), which inherently involves an associated reproducing kernel, and prove that every vv-RKBS belongs to such a pair. Our construction extends existing kernel 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 an integral 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 vv-RKBS. In all cases, we establish a representer theorem, showing that optimization over these function spaces recovers the corresponding neural architectures.
Functional Analysis (math.FA); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Machine Learning (stat.ML)
G.1.2; G.1.6; I.5.1; I.2.6
arXiv:2509.26371 [math.FA]
(or arXiv:2509.26371v3 [math.FA] for this version)
Full text under the publisher's open licence · 201 words
Related coverage
Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks
arXiv · Jun 17, 2026
Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks
arXiv · Jun 16, 2026
PIRAMID: Progress and Plans
LessWrong · Jul 27, 2026
Model Weight Profiles: Where Do the Parameters Go?
AMD · Sep 25, 2026