Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

Learning the Kernel Matrix with Semi-Definite Programming

Gert R. G. Lanckriet, Nello Cristianini, Peter Bartlett, Laurent El Ghaoui and Michael I. Jordan

EECS Department
University of California, Berkeley
Technical Report No. UCB/CSD-02-1206
2002

http://www.eecs.berkeley.edu/Pubs/TechRpts/2002/CSD-02-1206.pdf

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive definite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space -- classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semi-definite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -- using the labelled part of the data one can learn an embedding also for the unlabelled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method to learn the 2-norm soft margin parameter in support vector machines, solving another important open problem. Finally, the novel approach presented in the paper is supported by positive empirical results.


BibTeX citation:

@techreport{Lanckriet:CSD-02-1206,
    Author = {Lanckriet, Gert R. G. and Cristianini, Nello and Bartlett, Peter and El Ghaoui, Laurent and Jordan, Michael I.},
    Title = {Learning the Kernel Matrix with Semi-Definite Programming},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {2002},
    URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2002/5765.html},
    Number = {UCB/CSD-02-1206},
    Abstract = {Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive definite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space -- classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semi-definite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -- using the labelled part of the data one can learn an embedding also for the unlabelled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method to learn the 2-norm soft margin parameter in support vector machines, solving another important open problem. Finally, the novel approach presented in the paper is supported by positive empirical results.}
}

EndNote citation:

%0 Report
%A Lanckriet, Gert R. G.
%A Cristianini, Nello
%A Bartlett, Peter
%A El Ghaoui, Laurent
%A Jordan, Michael I.
%T Learning the Kernel Matrix with Semi-Definite Programming
%I EECS Department, University of California, Berkeley
%D 2002
%@ UCB/CSD-02-1206
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/2002/5765.html
%F Lanckriet:CSD-02-1206