On learning vector-valued functions. (English) Zbl 1092.93045

Micchelli, Charles A.; Pontil, Massimiliano

Neural Comput. 17, No. 1, 177-204 (2005).

Summary: 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 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.

MSC:

93E35	Stochastic learning and adaptive control
68T05	Learning and adaptive systems in artificial intelligence

Keywords:

Hilbert space

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