2025/09/30 by Sven Dummer, Tjeerd Jan Heeringa, Dummer, Sven +3
Computer Science · #Neural Networks and Applications #cs.AI #cs.LG #math.FA #stat.ML
paper · pdf · doi:10.48550/arxiv.2509.26371
openalex publication_date 2025/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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), ℝd-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 ℝd-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.