×

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

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DOI: 10.2307/1990404 · Zbl 0037.20701
[2] Bennett K.P., Optimization Methods and Software 3 pp 722– (1993)
[3] DOI: 10.1126/science.272.5270.1905
[4] DOI: 10.1111/1467-9868.00054
[5] DOI: 10.1111/j.1467-9868.2004.02054.x · Zbl 1046.62023
[6] DOI: 10.1162/15324430260185628 · Zbl 1037.68110
[7] DOI: 10.1023/A:1018946025316 · Zbl 0939.68098
[8] DOI: 10.1016/0024-3795(93)00232-O · Zbl 0852.43004
[9] DOI: 10.1109/5254.736001 · Zbl 05095844
[10] DOI: 10.1137/0716007 · Zbl 0399.65037
[11] DOI: 10.1162/15324430260185556 · Zbl 1021.68075
[12] Sejnowski T.J., Complex Systems 1 pp 145– (1987)
[13] DOI: 10.1109/34.735807
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.