# Interpolation and Learning with Scale Dependent Kernels

arXiv: https://arxiv.org/abs/2006.09984 · HTML: https://mlsanity.com/paper/2006.09984/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2020.

## Built on by (most cited first, 7 of 7)

- [PSD Representations for Effective Probability Models](https://mlsanity.com/paper/2106.16116/) (2021)
- [Spectral Analysis of the Neural Tangent Kernel for Deep Residual Networks](https://mlsanity.com/paper/2104.03093/) (2021)
- [Online nonparametric regression with Sobolev kernels](https://mlsanity.com/paper/2102.03594/) (2021)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [How many samples are needed to leverage smoothness?](https://mlsanity.com/paper/2305.16014/) (2023)
- [Impact of Computation in Integral Reinforcement Learning for Continuous-Time Control](https://mlsanity.com/paper/2402.17375/) (2024)
- [Interpolation with the polynomial kernels](https://mlsanity.com/paper/2212.07658/) (2022)

## Abstract (arXiv)

We study the learning properties of nonparametric ridge-less least squares. In particular, we consider the common case of estimators defined by scale dependent kernels, and focus on the role of the scale. These estimators interpolate the data and the scale can be shown to control their stability through the condition number. Our analysis shows that are different regimes depending on the interplay between the sample size, its dimensions, and the smoothness of the problem. Indeed, when the sample size is less than exponential in the data dimension, then the scale can be chosen so that the learning error decreases. As the sample size becomes larger, the overall error stop decreasing but interestingly the scale can be chosen in such a way that the variance due to noise remains bounded. Our analysis combines, probabilistic results with a number of analytic techniques from interpolation theory.