Interpolation and Learning with Scale Dependent Kernels

TL;DR

This paper analyzes how scale-dependent kernels influence the learning and stability of nonparametric ridge-less least squares estimators, revealing different regimes based on data size, dimension, and smoothness.

Contribution

It provides a theoretical analysis of the role of scale in kernel-based estimators, highlighting regimes where error decreases or variance remains bounded.

Findings

Error decreases when sample size is less than exponential in data dimension.

Scale can be chosen to keep noise variance bounded as data size grows.

Different regimes depend on sample size, dimension, and smoothness.

Abstract

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.