aiqbench analysis
This research advances Neural Operator theory by introducing a novel architecture that approximates nonlinear functionals on reproducing kernel Hilbert spaces using finite point evaluations rather than traditional integration-based expansions. The study provides rigorous theoretical guarantees, including universal approximation results and explicit bounds for specific kernels like Gaussian and Sobolev, which enhances the reliability of deep learning models for functional data such as time series and images. By establishing finite-sample generalization properties, this work bridges statistical learning theory with practical neural network design for complex functional regression tasks.
Neural Operators for Nonlinear Functionals on RKHS
arXiv:2403.12187v2 Announce Type: replace-cross Abstract: Motivated by the abundance of functional data, such as time series and images, we study the approximation and statistical learning of nonlinear functionals defined on reproducing kernel Hilbert spaces (RKHSs) using neural networks. By leveraging interpolating orthogonal projections in RKHSs, we use finitely many point evaluations in place of integration-based basis function expansions. This leads to a simpler and more flexible neural-network architecture that remains applicable even when the data domain or the underlying kernel is not explicitly known. We establish universal approximation results and derive explicit kernel-dependent approximation rates and parameter-complexity bounds for RKHSs induced by inverse multiquadric, Gaussian, and Sobolev kernels. We also apply our results to the regression maps arising in generalized functional linear models. Finally, we analyze the generalization properties of the resulting neural-network classes and establish finite-sample guarantees for learning nonlinear functionals.
Abstract:Motivated by the abundance of functional data, such as time series and images, we study the approximation and statistical learning of nonlinear functionals defined on reproducing kernel Hilbert spaces (RKHSs) using neural networks. By leveraging interpolating orthogonal projections in RKHSs, we use finitely many point evaluations in place of integration-based basis function expansions. This leads to a simpler and more flexible neural-network architecture that remains applicable even when the data domain or the underlying kernel is not explicitly known. We establish universal approximation results and derive explicit kernel-dependent approximation rates and parameter-complexity bounds for RKHSs induced by inverse multiquadric, Gaussian, and Sobolev kernels. We also apply our results to the regression maps arising in generalized functional linear models. Finally, we analyze the generalization properties of the resulting neural-network classes and establish finite-sample guarantees for learning nonlinear functionals.
Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)
arXiv:2403.12187 [stat.ML]
(or arXiv:2403.12187v2 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2403.12187 Focus to learn more arXiv-issued DOI via DataCite
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 235 words
Related coverage
Learning Lyapunov Operators for Nonlinear Systems
arXiv · Sep 17, 2026
Quantum chip predicts time series with a feedback loop
Quantum Zeitgeist · Sep 27, 2026
Variational Boosting for Physics-Informed Neural Networks
arXiv · Jul 28, 2026
KernelOnet: An Interpretable Neural Operator Based on Kernel Functions
arXiv · Sep 30, 2026