On the kernel learning problem

• URL: http://arxiv.org/abs/2502.11665v1
• Date: Mon, 17 Feb 2025 10:54:01 GMT
• Title: On the kernel learning problem
• Authors: Yang Li, Feng Ruan,
• Abstract summary: kernel ridge regression problem aims to find the best fit for the output $Y$ as a function of the input data $Xin mathbbRd$.<n>We consider a generalization of the kernel ridge regression problem, by introducing an extra matrix parameter $U$.<n>This naturally leads to a nonlinear variational problem to optimize the choice of $U$.
• Score: 4.917649865600782
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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