## Learning the Kernel Matrix with Semi-Definite Programming

Gert Lanckriet, Nello Cristianini^{1}, and Peter Bartlett^{2}

(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.

^{1}Professor, Dept. of Statistics, UC Davis

^{2}Visiting Professor, A.N.U., Canberra

More information (http://robotics.eecs.berkeley.edu/~gert/) or

Send mail to the author : (gert@eecs.berkeley.edu)

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