  We study the problem of structured output learning from a regression
perspective. We first provide a general formulation of the kernel dependency
estimation (KDE) problem using operator-valued kernels. We show that some of
the existing formulations of this problem are special cases of our framework.
We then propose a covariance-based operator-valued kernel that allows us to
take into account the structure of the kernel feature space. This kernel
operates on the output space and encodes the interactions between the outputs
without any reference to the input space. To address this issue, we introduce a
variant of our KDE method based on the conditional covariance operator that in
addition to the correlation between the outputs takes into account the effects
of the input variables. Finally, we evaluate the performance of our KDE
approach using both covariance and conditional covariance kernels on two
structured output problems, and compare it to the state-of-the-art kernel-based
structured output regression methods.
