Kernel Interpolation in Sobolev Spaces Is Not Consistent in Low Dimensions
Abstract
We consider kernel ridgeless ridge regression with kernels whose associated RKHS is a Sobolev space $H^s$. We show for $d/2<s<3d/4$ that interpolation is not consistent in fixed dimension extending earlier results for the Laplace kernel in odd dimensions and underlining again that benign overfitting is rare in low dimensions. The proof proceeds by deriving sharp bounds on the spectrum of random kernel matrices using results from the theory of radial basis functions which might be of independent interest.
Cite
Text
Buchholz. "Kernel Interpolation in Sobolev Spaces Is Not Consistent in Low Dimensions." Conference on Learning Theory, 2022.Markdown
[Buchholz. "Kernel Interpolation in Sobolev Spaces Is Not Consistent in Low Dimensions." Conference on Learning Theory, 2022.](https://mlanthology.org/colt/2022/buchholz2022colt-kernel/)BibTeX
@inproceedings{buchholz2022colt-kernel,
title = {{Kernel Interpolation in Sobolev Spaces Is Not Consistent in Low Dimensions}},
author = {Buchholz, Simon},
booktitle = {Conference on Learning Theory},
year = {2022},
pages = {3410-3440},
volume = {178},
url = {https://mlanthology.org/colt/2022/buchholz2022colt-kernel/}
}