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#### Maximal Margin Classification for Metric Spaces

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##### Citation

Hein, M., Bousquet, O., & Schölkopf, B. (2005). Maximal Margin Classification for
Metric Spaces.* Journal of Computer and System Sciences,* *71*(3),
333-359. doi:10.1016/j.jcss.2004.10.013.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-D3F9-2

##### Abstract

In order to apply the maximum margin method in arbitrary metric
spaces, we suggest to embed the metric space into a Banach or
Hilbert space and to perform linear classification in this space.
We propose several embeddings and recall that an isometric embedding
in a Banach space is always possible while an isometric embedding in
a Hilbert space is only possible for certain metric spaces. As a
result, we obtain a general maximum margin classification
algorithm for arbitrary metric spaces (whose solution is
approximated by an algorithm of Graepel.
Interestingly enough, the embedding approach, when applied to a metric
which can be embedded into a Hilbert space, yields the SVM
algorithm, which emphasizes the fact that its solution depends on
the metric and not on the kernel. Furthermore we give upper bounds
of the capacity of the function classes corresponding to both
embeddings in terms of Rademacher averages. Finally we compare the
capacities of these function classes directly.