Refinement of Operator-valued Reproducing Kernels

TL;DR

This paper introduces a method to refine operator-valued reproducing kernels, enabling better adaptation in multi-task learning by expanding the associated Hilbert space, supported by theoretical characterizations and numerical simulations.

Contribution

It proposes a novel refinement kernel construction for operator-valued kernels, with characterizations, examples, and analysis of properties during refinement.

Findings

Refinement kernels can effectively update existing kernels in multi-task learning.

Numerical simulations demonstrate the method's ability to address underfitting and overfitting.

Various classes of kernels, including translation-invariant and Hessian kernels, are refined successfully.

Abstract

This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace. The study is motivated from the need of updating the current operator-valued reproducing kernel in multi-task learning when underfitting or overfitting occurs. Numerical simulations confirm that the established refinement kernel method is able to meet this need. Various characterizations are provided based on feature maps and vector-valued integral representations of operator-valued reproducing kernels. Concrete examples of refining translation invariant and finite Hilbert-Schmidt operator-valued reproducing kernels are provided. Other examples include refinement of Hessian of scalar-valued translation-invariant kernels and transformation kernels. Existence and properties of operator-valued reproducing kernels preserved during the refinement process are also investigated.