Finite rank kernels for multi-task learning

Authors:
Jianqiang Liu;Charles A. Micchelli;Rui Wang;Yuesheng Xu
Affiliations:
Department of Mathematics and Computer Science, Ningxia University, Yinchuan, People's Republic of China 750021;Department of Mathematics and Statistics, State University of New York, The University at Albany, Albany, USA 12222 and Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong;College of Mathematics, Jilin University, Changchun, People's Republic of China 130012;Department of Mathematics, Syracuse University, Syracuse, USA 13244 and Guangdong Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, People's Republic of China 510275
Venue:
Advances in Computational Mathematics
Year:
2013

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Abstract

Motivated by the importance of kernel-based methods for multi-task learning, we provide here a complete characterization of multi-task finite rank kernels in terms of the positivity of what we call its associated characteristic operator. Consequently, we are led to establishing that every continuous multi-task kernel, defined on a cube in an Euclidean space, not only can be uniformly approximated by multi-task polynomial kernels, but also can be extended as a multi-task kernel to all of the Euclidean space. Finally, we discuss the interpolation of multi-task kernels by multi-task finite rank kernels.