title: On learning vector-valued functions.
abstract: 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.
authors.affiliation: Department of Mathematics and Statistics, State University of New York, University at Albany, Albany, NY, 12222, USA. charles_Micchelli@hotmail.com
authors.forename: Charles A
authors.initials: CA
authors.lastname: Micchelli
authors.forename: Massimiliano
authors.initials: M
authors.lastname: Pontil
mesh_terms: D000465:Algorithms; D001185:Artificial Intelligence; D008432:Mathematical Computing; D008962:Models, Theoretical; D016571:Neural Networks, Computer; D012044:Regression Analysis