Kernel methods for vector output

Kernel methods are a well-established tool to analyze the relationship between input data and the corresponding output of a function. Kernels encapsulate the properties of functions in a computationally efficient way and allow algorithms to easily swap functions of varying complexity.

In typical machine learning algorithms, these functions produce a scalar output. Recent development of kernel methods for functions with vector-valued output is due, at least in part, to interest in simultaneously solving related problems. Kernels which capture the relationship between the problems allow them to borrow strength from each other. Algorithms of this type include multi-task learning (also called multi-output learning or vector-valued learning), transfer learning, and co-kriging. Multi-label classification can be interpreted as mapping inputs to (binary) coding vectors with length equal to the number of classes.

In gaussian processes, kernels are called covariance functions. Multiple-output functions correspond to considering multiple processes. See Bayesian interpretation of regularization for the connection between the two perspectives.