On the kernel learning problem

TL;DR

This paper explores a generalized kernel ridge regression framework that incorporates an additional matrix parameter to better detect data features and scales, addressing multiscale structures in the data.

Contribution

It introduces a novel variational formulation for kernel ridge regression with a matrix parameter, analyzing its mathematical properties and effectiveness for multiscale data.

Findings

Mathematical analysis of the variational problem

Insights into multiscale data behavior

Potential improvements in kernel regression efficiency

Abstract

The classical kernel ridge regression problem aims to find the best fit for the output $Y$ as a function of the input data $X\in \mathbb{R}^d$, with a fixed choice of regularization term imposed by a given choice of a reproducing kernel Hilbert space, such as a Sobolev space. Here we consider a generalization of the kernel ridge regression problem, by introducing an extra matrix parameter $U$, which aims to detect the scale parameters and the feature variables in the data, and thereby improve the efficiency of kernel ridge regression. This naturally leads to a nonlinear variational problem to optimize the choice of $U$. We study various foundational mathematical aspects of this variational problem, and in particular how this behaves in the presence of multiscale structures in the data.