Learning the Kernel Matrix with Semi-Definite Programming

Gert Lanckriet, Nello Cristianini1, and Peter Bartlett2
(Professors Laurent El Ghaoui and Michael I. Jordan)
(NSF) IIS-9988642 and (ONR) MURI N00014-00-1-0637

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 project 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 labeled part of the data one can learn an embedding also for the unlabeled 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. Finally, the novel approach presented in the project is supported by positive empirical results.

1Professor, Dept. of Statistics, UC Davis
2Visiting Professor, A.N.U., Canberra

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