  We consider the problem of operator-valued kernel learning and investigate
the possibility of going beyond the well-known separable kernels. Borrowing
tools and concepts from the field of quantum computing, such as partial trace
and entanglement, we propose a new view on operator-valued kernels and define a
general family of kernels that encompasses previously known operator-valued
kernels, including separable and transformable kernels. Within this framework,
we introduce another novel class of operator-valued kernels called entangled
kernels that are not separable. We propose an efficient two-step algorithm for
this framework, where the entangled kernel is learned based on a novel
extension of kernel alignment to operator-valued kernels. We illustrate our
algorithm with an application to supervised dimensionality reduction, and
demonstrate its effectiveness with both artificial and real data for
multi-output regression.
