Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operators

Mike Nguyen, Nicole Mücke

Abstract:

In this work, we investigate the generalization properties of random feature methods. Our analysis extends prior
results for Tikhonov regularization to a broad class of spectral regularization techniques and further generalizes
the setting to operator-valued kernels. This unified framework enables, for the first time, a rigorous theoretical
analysis of neural operators and neural networks through the lens of the Neural Tangent Kernel (NTK). In
particular, it allows us to establish optimal learning rates and provides a good understanding of how many
neurons are required to achieve a given accuracy. Furthermore, we establish minimax rates in the well-specified
case and also in the misspecified case, where the target is not contained in the reproducing kernel Hilbert space.
These results sharpen and complete earlier findings for specific kernel algorithms.

Code Dataset Promise: No

Format Confirmation: I agree that I have read and followed the formatting instructions for the camera ready version.

Submission Number: 643