Regularised Least-Squares Regression with Infinite-Dimensional Output Space

TL;DR

This paper explores theoretical aspects of vector-valued RKHS regression with infinite-dimensional outputs, emphasizing minimal assumptions and spectral operator techniques to derive learning results.

Contribution

It provides new theoretical insights into vector-valued RKHS regression with non-compact input spaces and infinite-dimensional outputs using spectral theory.

Findings

Results obtained with minimal assumptions

Use of spectral theory for non-compact operators

Focus on theoretical learning bounds

Abstract

This short technical report presents some learning theory results on vector-valued reproducing kernel Hilbert space (RKHS) regression, where the input space is allowed to be non-compact and the output space is a (possibly infinite-dimensional) Hilbert space. Our approach is based on the integral operator technique using spectral theory for non-compact operators. We place a particular emphasis on obtaining results with as few assumptions as possible; as such we only use Chebyshev's inequality, and no effort is made to obtain the best rates or constants.