On the kernel learning problem
This work addresses a foundational problem in kernel learning for researchers in machine learning and statistics, but it appears incremental as it builds on classical kernel ridge regression.
The authors tackled the kernel ridge regression problem by introducing an extra matrix parameter to detect scale and feature variables, aiming to improve efficiency, and they studied its mathematical aspects in multiscale data contexts.
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.