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.


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


Hilbert space
Full Text: DOI


[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
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