{"ID":2862850,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.26371","arxiv_id":"2509.26371","title":"Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators","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 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.","short_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 underst...","url_abs":"https://arxiv.org/abs/2509.26371","url_pdf":"https://arxiv.org/pdf/2509.26371v2","authors":"[\"Sven Dummer\",\"Tjeerd Jan Heeringa\",\"José A. Iglesias\"]","published":"2025-09-30T15:06:24Z","proceeding":"math.FA","tasks":"[\"math.FA\",\"cs.AI\",\"cs.LG\",\"stat.ML\"]","methods":"[]","has_code":false}