  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.
