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Learning bounds for kernel regression using effective data dimensionality.

Tong Zhang1

  • 1IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA. tzhang@watson.ibm.com

Neural Computation
|July 5, 2005
PubMed
Summary
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Kernel methods embed data into high-dimensional spaces but possess a small effective dimension. This research defines a scale-sensitive dimension to bound generalization performance in kernel regression, showing optimal convergence rates.

Area of Science:

  • Machine Learning
  • Statistical Learning Theory

Background:

  • Kernel methods map finite data to infinite-dimensional feature spaces, achieving good generalization.
  • This success is sometimes misinterpreted as avoiding the curse of dimensionality.

Purpose of the Study:

  • To demonstrate that kernel embeddings have a small effective dimensionality, not infinite.
  • To introduce a scale-sensitive effective dimension for kernel representations.
  • To derive generalization bounds for kernel regression based on this dimension.

Main Methods:

  • Algebraic definition of a scale-sensitive effective dimension.
  • Derivation of upper bounds on generalization error for kernel regression.
  • Analysis of convergence rates for kernel learning algorithms.

Related Experiment Videos

Main Results:

  • Kernel embeddings, despite infinite dimensionality, possess a small, finite effective dimension.
  • The derived bounds on generalization performance are tight.
  • Optimal convergence rates are achieved for kernel regression under specific conditions.

Conclusions:

  • The effective dimensionality, not the feature space dimension, governs kernel machine complexity.
  • The scale-sensitive effective dimension provides a principled way to analyze kernel generalization.
  • This work clarifies the theoretical underpinnings of kernel method success in machine learning.