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Input space versus feature space in kernel-based methods




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.

Author(s): Schölkopf, B. and Mika, S. and Burges, CJC. and Knirsch, P. and Müller, K-R. and Rätsch, G. and Smola, AJ.
Journal: IEEE Transactions On Neural Networks
Volume: 10
Number (issue): 5
Pages: 1000-1017
Year: 1999
Month: September
Day: 0

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

Digital: 0
DOI: 10.1109/72.788641
Organization: Max-Planck-Gesellschaft
School: Biologische Kybernetik

Links: Web


  title = {Input space versus feature space in kernel-based methods },
  author = {Sch{\"o}lkopf, B. and Mika, S. and Burges, CJC. and Knirsch, P. and M{\"u}ller, K-R. and R{\"a}tsch, G. and Smola, AJ.},
  journal = {IEEE Transactions On Neural Networks},
  volume = {10},
  number = {5},
  pages = {1000-1017},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  month = sep,
  year = {1999},
  month_numeric = {9}