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Sobolev norm inconsistency of kernel interpolation

arXiv 2504.20617 ↗First posted 29 Apr 2025stat.ML cs.LGcited by 0

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Abstract (arXiv)

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 L2 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.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 26 of its 37 references that are in the map. First posted on arXiv 29 Apr 2025.

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References · 26

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,621
  2. 2016Understanding deep learning requires rethinking generalizationcited by 1,674
  3. 2018Reconciling modern machine learning practice and the bias-variance trade-offcited by 927
  4. 2019Benign Overfitting in Linear Regressioncited by 549
  5. 2019Surprises in High-Dimensional Ridgeless Least Squares Interpolationcited by 513
  6. 2021Deep learning: a statistical viewpointcited by 240
  7. 2019Uniform convergence may be unable to explain generalization in deep learningcited by 224
  8. 2018Just Interpolate: Kernel "Ridgeless" Regression Can Generalizecited by 190
  9. 2019On the Inductive Bias of Neural Tangent Kernelscited by 177
  10. 2021Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolationcited by 132
  11. 2017Sobolev Norm Learning Rates for Regularized Least-Squares Algorithmcited by 114
  12. 2021Generalization error of random features and kernel methods: hypercontractivity and kernel matrix concentrationcited by 110
14 more
  1. 2022Benign Overfitting in Two-layer Convolutional Neural Networkscited by 91
  2. 2016Optimal Rates For Regularization Of Statistical Inverse Learning Problemscited by 68
  3. 2018Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spacescited by 59
  4. 2021Understanding neural networks with reproducing kernel Banach spacescited by 50
  5. 2018Consistency of Interpolation with Laplace Kernels is a High-Dimensional Phenomenoncited by 44
  6. 2016Distributed learning with regularized least squarescited by 42
  7. 2021Near-Minimax Optimal Estimation With Shallow ReLU Neural Networkscited by 41
  8. 2019On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructionscited by 29
  9. 2023On the Optimality of Misspecified Spectral Algorithmscited by 28
  10. 2023Mind the spikes: Benign overfitting of kernels and neural networks in fixed dimensioncited by 26
  11. 2023Nonparametric regression using over-parameterized shallow ReLU neural networkscited by 24
  12. 2023Kernel interpolation generalizes poorlycited by 16
  13. 2022On the Inconsistency of Kernel Ridgeless Regression in Fixed Dimensionscited by 13
  14. 2020Kernel Interpolation of High Dimensional Scattered Datacited by 6

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