# Consistency of Interpolation with Laplace Kernels is a High-Dimensional Phenomenon

arXiv: https://arxiv.org/abs/1812.11167 · HTML: https://mlsanity.com/paper/1812.11167/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Dec 2018.

## Built on by (most cited first, 30 of 44)

- [Spectrum of inner-product kernel matrices in the polynomial regime and multiple descent phenomenon in kernel ridge regression](https://mlsanity.com/paper/2204.10425/) (2022)
- [Benign, Tempered, or Catastrophic: A Taxonomy of Overfitting](https://mlsanity.com/paper/2207.06569/) (2022)
- [On the Eigenvalue Decay Rates of a Class of Neural-Network Related Kernel Functions Defined on General Domains](https://mlsanity.com/paper/2305.02657/) (2023)
- [Towards an Understanding of Benign Overfitting in Neural Networks](https://mlsanity.com/paper/2106.03212/) (2021)
- [Generalization Ability of Wide Neural Networks on ℝ](https://mlsanity.com/paper/2302.05933/) (2023)
- [Benign Overfitting in Linear Classifiers and Leaky ReLU Networks from KKT Conditions for Margin Maximization](https://mlsanity.com/paper/2303.01462/) (2023)
- [Generalization in Kernel Regression Under Realistic Assumptions](https://mlsanity.com/paper/2312.15995/) (2023)
- [Mind the spikes: Benign overfitting of kernels and neural networks in fixed dimension](https://mlsanity.com/paper/2305.14077/) (2023)
- [Nonparametric regression using over-parameterized shallow ReLU neural networks](https://mlsanity.com/paper/2306.08321/) (2023)
- [Six Lectures on Linearized Neural Networks](https://mlsanity.com/paper/2308.13431/) (2023)
- [Sharp Asymptotics of Kernel Ridge Regression Beyond the Linear Regime](https://mlsanity.com/paper/2205.06798/) (2022)
- [On the Asymptotic Learning Curves of Kernel Ridge Regression under Power-law Decay](https://mlsanity.com/paper/2309.13337/) (2023)
- [Kernel interpolation generalizes poorly](https://mlsanity.com/paper/2303.15809/) (2023)
- [From Tempered to Benign Overfitting in ReLU Neural Networks](https://mlsanity.com/paper/2305.15141/) (2023)
- [Harmless interpolation in regression and classification with structured features](https://mlsanity.com/paper/2111.05198/) (2021)
- [On the Inconsistency of Kernel Ridgeless Regression in Fixed Dimensions](https://mlsanity.com/paper/2205.13525/) (2022)
- [Wide and Deep Neural Networks Achieve Optimality for Classification](https://mlsanity.com/paper/2204.14126/) (2022)
- [Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay](https://mlsanity.com/paper/2401.01599/) (2024)
- [Strong inductive biases provably prevent harmless interpolation](https://mlsanity.com/paper/2301.07605/) (2023)
- [The phase diagram of kernel interpolation in large dimensions](https://mlsanity.com/paper/2404.12597/) (2024)
- [Overfitting Behaviour of Gaussian Kernel Ridgeless Regression: Varying Bandwidth or Dimensionality](https://mlsanity.com/paper/2409.03891/) (2024)
- [Nonparametric Regression with Shallow Overparameterized Neural Networks Trained by GD with Early Stopping](https://mlsanity.com/paper/2107.05341/) (2021)
- [Learning Lipschitz Functions by GD-trained Shallow Overparameterized ReLU Neural Networks](https://mlsanity.com/paper/2212.13848/) (2022)
- [Generalization Ability of Wide Residual Networks](https://mlsanity.com/paper/2305.18506/) (2023)
- [Measuring Complexity of Learning Schemes Using Hessian-Schatten Total Variation](https://mlsanity.com/paper/2112.06209/) (2021)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [On the Consistency of Kernel Methods with Dependent Observations](https://mlsanity.com/paper/2406.06101/) (2024)
- [Benign overfitting and adaptive nonparametric regression](https://mlsanity.com/paper/2206.13347/) (2022)
- [Transfer Learning for Benign Overfitting in High-Dimensional Linear Regression](https://mlsanity.com/paper/2510.15337/) (2025)
- [Sobolev Space Regularised Pre Density Models](https://mlsanity.com/paper/2307.13763/) (2023)

## Abstract (arXiv)

We show that minimum-norm interpolation in the Reproducing Kernel Hilbert Space corresponding to the Laplace kernel is not consistent if input dimension is constant. The lower bound holds for any choice of kernel bandwidth, even if selected based on data. The result supports the empirical observation that minimum-norm interpolation (that is, exact fit to training data) in RKHS generalizes well for some high-dimensional datasets, but not for low-dimensional ones.