On relevant dimensions in kernel feature spaces

Mikio L. Braun, Joachim M. Buhmann, Klaus Robert Müller

Research output: Contribution to journalArticlepeer-review

103 Citations (Scopus)


We show that the relevant information of a supervised learning problem is contained up to negligible error in a finite number of leading kernel PCA components if the kernel matches the underlying learning problem in the sense that it can asymptotically represent the function to be learned and is sufficiently smooth. Thus, kernels do not only transform data sets such that good generalization can be achieved using only linear discriminant functions, but this transformation is also performed in a manner which makes economical use of feature space dimensions. In the best case, kernels provide efficient implicit representations of the data for supervised learning problems. Practically, we propose an algorithm which enables us to recover the number of leading kernel PCA components relevant for good classification. Our algorithm can therefore be applied (1) to analyze the interplay of data set and kernel in a geometric fashion, (2) to aid in model selection, and (3) to denoise in feature space in order to yield better classification results.

Original languageEnglish
Pages (from-to)1875-1908
Number of pages34
JournalJournal of Machine Learning Research
Publication statusPublished - 2008 Aug
Externally publishedYes


  • Dimension reduction
  • Effective dimensionality
  • Feature space
  • Kernel methods

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Statistics and Probability
  • Artificial Intelligence


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