On learning vector-valued functions

Neural Comput. 2005 Jan;17(1):177-204. doi: 10.1162/0899766052530802.


In this letter, we provide a study of learning in a Hilbert space of vectorvalued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a reproducing kernel Hilbert space of functions whose values lie in Y. In this setting, we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory. We consider specific examples of such functionals corresponding to multiple-output regularization networks and support vector machines, for both regression and classification. Finally, we provide classes of operator-valued kernels of the dot product and translation-invariant type.

Publication types

  • Research Support, U.S. Gov't, Non-P.H.S.

MeSH terms

  • Algorithms*
  • Artificial Intelligence*
  • Mathematical Computing*
  • Models, Theoretical
  • Neural Networks, Computer*
  • Regression Analysis