Sobolev norm inconsistency of kernel interpolation

We study the consistency of minimum-norm interpolation in reproducing kernel Hilbert spaces corresponding to bounded kernels. Our main result give lower bounds for the generalization error of the kernel interpolation measured in a continuous scale of norms that interpolate between $L^2$ and the hypothesis space. These lower bounds imply that kernel interpolation is always inconsistent, when the smoothness index of the norm is larger than a constant that depends only on the embedding index of the hypothesis space and the decay rate of the eigenvalues.

@article{yang2025_2504.20617,
  title={ Sobolev norm inconsistency of kernel interpolation },
  author={ Yunfei Yang },
  journal={arXiv preprint arXiv:2504.20617},
  year={ 2025 }
}