A Variational Analysis of Kernel Learning with Learnable Linear Transformations

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, including its Euler-Lagrange equation, continuity and first variation, limiting behavior under degenerate or diverging transformations, and the structure of its local minimizers. Particular attention is given to two data-distribution settings, namely multi-scale and multi-index models, where the learned transformation $U$ encodes intrinsic scale parameters and the essential low-dimensional feature variables, respectively.

68 pages, revised version