PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

When Is There a Representer Theorem? Vector versus Matrix Regularizers
Andreas Argyriou, Charles Micchelli and Massimiliano Pontil
Journal of Machine Learning Research 2008.


We consider a general class of regularization methods which learn a vector of parameters on the basis of linear measurements. It is well known that if the regularizer is a nondecreasing function of the inner product then the learned vector is a linear combination of the input data. This result, known as the representer theorem, is at the basis of kernel-based methods in machine learning. In this paper, we prove the necessity of the above condition, thereby completing the characterization of kernel methods based on regularization. We further extend our analysis to regularization methods which learn a matrix, a problem which is motivated by the application to multi-task learning. In this context, we study a more general representer theorem, which holds for a larger class of regularizers. We provide a necessary and sufficient condition for these class of matrix regularizers and highlight them with some concrete examples of practical importance. Our analysis uses basic principles from matrix theory, especially the useful notion of matrix nondecreasing function.

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EPrint Type:Article
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Computational, Information-Theoretic Learning with Statistics
Learning/Statistics & Optimisation
ID Code:4732
Deposited By:Andreas Argyriou
Deposited On:24 March 2009