# Kernels for Vector-Valued Functions: A Review

Mauricio A. Álvarez, Universidad Tecnológica de Pereira, Colombia
Lorenzo Rosasco, University of Genoa
Neil D. Lawrence, University of Sheffield

Foundations and Trends in Machine Learning 4, pp 195-266

#### Abstract

Kernel methods are among the most popular techniques in machine learning. From a regularization perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the regularization functional through the notion of reproducing kernel Hilbert spaces. From a probabilistic perspec- tive they are the key in the context of Gaussian processes, where the kernel function is known as the covariance function. Traditionally, kernel methods have been used in supervised learning problems with scalar outputs and indeed there has been a considerable amount of work devoted to designing and learning kernels. More recently there has been an increasing interest in methods that deal with multiple outputs, motivated partially by frameworks like multitask learning. In this monograph, we review different methods to design or learn valid kernel functions for multiple outputs, paying particular attention to the connection between probabilistic and functional methods.

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 %T Kernels for Vector-Valued Functions: A Review %A Mauricio A. Álvarez and Lorenzo Rosasco and Neil D. Lawrence %B %C Foundations and Trends in Machine Learning %D %F alvarez-vector12 %J Foundations and Trends in Machine Learning %P 195--266 %R 10.1561/2200000036 %U http://inverseprobability.com/publications/alvarez-vector12.html %V 4 %N 3 %X Kernel methods are among the most popular techniques in machine learning. From a regularization perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the regularization functional through the notion of reproducing kernel Hilbert spaces. From a probabilistic perspec- tive they are the key in the context of Gaussian processes, where the kernel function is known as the covariance function. Traditionally, kernel methods have been used in supervised learning problems with scalar outputs and indeed there has been a considerable amount of work devoted to designing and learning kernels. More recently there has been an increasing interest in methods that deal with multiple outputs, motivated partially by frameworks like multitask learning. In this monograph, we review different methods to design or learn valid kernel functions for multiple outputs, paying particular attention to the connection between probabilistic and functional methods. 
 TY - CPAPER TI - Kernels for Vector-Valued Functions: A Review AU - Mauricio A. Álvarez AU - Lorenzo Rosasco AU - Neil D. Lawrence PY - 2012/01/01 DA - 2012/01/01 ID - alvarez-vector12 SP - 195 EP - 266 DO - 10.1561/2200000036 L1 - ftp://ftp.dcs.shef.ac.uk/home/neil/2200000036.pdf UR - http://inverseprobability.com/publications/alvarez-vector12.html AB - Kernel methods are among the most popular techniques in machine learning. From a regularization perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the regularization functional through the notion of reproducing kernel Hilbert spaces. From a probabilistic perspec- tive they are the key in the context of Gaussian processes, where the kernel function is known as the covariance function. Traditionally, kernel methods have been used in supervised learning problems with scalar outputs and indeed there has been a considerable amount of work devoted to designing and learning kernels. More recently there has been an increasing interest in methods that deal with multiple outputs, motivated partially by frameworks like multitask learning. In this monograph, we review different methods to design or learn valid kernel functions for multiple outputs, paying particular attention to the connection between probabilistic and functional methods. ER - 
 Álvarez, M.A., Rosasco, L. & Lawrence, N.D.. (2012). Kernels for Vector-Valued Functions: A Review. Foundations and Trends in Machine Learning 4(3):195-266