Input space versus feature space in kernel-based methods

Bernhard Schölkopf, Sebastian Mika, Chris J.C. Burges, Philipp Knirsch, Klaus Robert Müller, Gunnar Rätsch, Alexander J. Smola

Research output: Contribution to journalArticlepeer-review

1042 Citations (Scopus)


This paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector (SV) kernel functions. We first discuss the geometry of feature space. In particular, we review what is known about the shape of the image of input space under the feature space map, and how this influences the capacity of SV methods. Following this, we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel, using the example of the class of inhomogeneous polynomial kernels, which are often used in SV pattern recognition. We then discuss the connection between feature space and input space by dealing with the question of how one can, given some vector in feature space, find a preimage (exact or approximate) in input space. We describe algorithms to tackle this issue, and show their utility in two applications of kernel methods. First, we use it to reduce the computational complexity of SV decision functions; second, we combine it with the Kernel PCA algorithm, thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real-world data.

Original languageEnglish
Pages (from-to)1000-1017
Number of pages18
JournalIEEE Transactions on Neural Networks
Issue number5
Publication statusPublished - 1999

Bibliographical note

Funding Information:
Manuscript received January 23, 1999; revised May 16, 1999. Part of this work was done while P. Knirsch was with Bell Labs and B. Schölkopf and A. J. Smola were with the Department of Engineering, Australian National University, Canberra. This work was supported by the ARC and the DFG under Grant Ja 379/52,71,91. B. Schölkopf was with GMD FIRST, 12489 Berlin, Germany. He is now with Mocrosoft Research Ltd., Cambridge CB2, U.K. S. Mika, K.-R. Müller, G. Rätsch, and A. J. Smola are with GMD FIRST, 12489 Berlin, Germany. C. J. C. Burges is with Bell Laboratories, Holmdel NJ, USA. P. Knirsch is with the Max-Planck-Institut für biologische Kybernetik, 72076 Tübingen, Germany. Publisher Item Identifier S 1045-9227(99)07268-9.

ASJC Scopus subject areas

  • Software
  • Computer Science Applications
  • Computer Networks and Communications
  • Artificial Intelligence


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