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Maximal Margin Classification for Metric Spaces




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.

Author(s): Hein, M. and Bousquet, O. and Schölkopf, B.
Journal: Journal of Computer and System Sciences
Volume: 71
Number (issue): 3
Pages: 333-359
Year: 2005
Month: October
Day: 0

Department(s): Empirical Inference
Bibtex Type: Article (article)

Digital: 0
DOI: 10.1016/j.jcss.2004.10.013
Language: en
Organization: Max-Planck-Gesellschaft
School: Biologische Kybernetik

Links: PDF


  title = {Maximal Margin Classification for Metric Spaces},
  author = {Hein, M. and Bousquet, O. and Sch{\"o}lkopf, B.},
  journal = {Journal of Computer and System Sciences},
  volume = {71},
  number = {3},
  pages = {333-359},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  month = oct,
  year = {2005},
  month_numeric = {10}