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Neural Comput. 2005 Jan;17(1):177-204.

On learning vector-valued functions.

Author information

1
Department of Mathematics and Statistics, State University of New York, University at Albany, Albany, NY, 12222, USA. charles_Micchelli@hotmail.com

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.

PMID:
15563752
DOI:
10.1162/0899766052530802
[Indexed for MEDLINE]

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