A Method to Construct the Mapping to the Feature Space for the Dot Product Kernels (original) (raw)

Abstract

Dot product kernels are a class of important kernel in the theory of support vector machine. This paper develops a method to construct the mapping that map the original data set into the high dimensional feature space, on which the inner product is defined by a dot product kernel. Our method can also be applied to the Gaussian kernels. Via this mapping, the structure of features in the feature space is easy to be observed, and the linear separability of data sets in the feature space is studied. We obtain that any two finite sets of data with empty overlap in the original space will become linearly separable in an infinite dimensional feature space, and a sufficient and necessary condition is also developed for two infinite sets of data in the original data space being linearly separable in the feature space, this condition can be applied to examine the existences and uniqueness of the hyperplane which can separate all the possible inputs correctly.

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Authors and Affiliations

  1. Department of Mathematics and Physics, North China Electric Power University, 102206, Beijing, P.R. China
    Degang Chen
  2. Department of Mathematics and Computer Science, Hebei University, Baoding, Hebei, 071002, P.R. China
    Qiang He, Chunru Dong & Xizhao Wang

Authors

  1. Degang Chen
  2. Qiang He
  3. Chunru Dong
  4. Xizhao Wang

Editor information

Editors and Affiliations

  1. Department of Computing, Hong Kong Polytechnic University, P.O. Box, Hong Kong, China
    Daniel S. Yeung
  2. School of Creative Media, City University of Hong Kong,, China
    Zhi-Qiang Liu
  3. Department of Mathematics and Computer Science, Hebei University, 071002, Baoding, Hebei, P.R. China
    Xi-Zhao Wang
  4. School of Electrical and Information Engineering, University of Sydney, 2006, NSW, Australia
    Hong Yan

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© 2006 Springer-Verlag Berlin Heidelberg

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Chen, D., He, Q., Dong, C., Wang, X. (2006). A Method to Construct the Mapping to the Feature Space for the Dot Product Kernels. In: Yeung, D.S., Liu, ZQ., Wang, XZ., Yan, H. (eds) Advances in Machine Learning and Cybernetics. Lecture Notes in Computer Science(), vol 3930. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11739685\_96

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