Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators
This provides a theoretical framework for analyzing vector-valued neural networks and operators, which is incremental as it extends existing scalar-valued RKBS theory to more complex models.
The paper tackled the gap in understanding vector-valued neural networks and neural operators within reproducing kernel Banach spaces (RKBS) by developing a general definition of vector-valued RKBS, and showed that shallow R^d-valued neural networks and neural operators like DeepONet and Hypernetwork belong to such spaces, with a Representer Theorem linking optimization to these architectures.
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.