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On Learning Vector-Valued Functions

charles_Micchelli{at}hotmail.com, Department of Mathematics and Statistics, State University of New York, University at Albany, Albany, NY, 12222, U.S.A.

Massimiliano Pontil

m.pontil{at}cs.uclac.uk, Department of Computer Sciences, University College London, London WC1E, England, UK

In this letter, we provide a study of learning in a Hilbert space of vector-valued 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 to be a Hilbert space, and we consider a reproducing kernel Hilbert space of functions whose values lie in . 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.

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