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A loss function approach to model selection in nonlinear principal components.

Andrew R. Webb1

  • 1Defence Evaluation and Research Agency, St Andrews Road, Malvern, UK

Neural Networks : the Official Journal of the International Neural Network Society
|March 29, 2003
PubMed
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This study models nonlinear principal component analysis using kernel functions. An alternating least squares algorithm minimizes a loss function for parameter estimation and model selection via cross-validation.

Area of Science:

  • Machine Learning
  • Statistical Modeling
  • Dimensionality Reduction

Background:

  • Nonlinear Principal Component Analysis (NLPCA) is crucial for capturing complex data structures.
  • Traditional methods often struggle with high-dimensional or nonlinear data.
  • Kernel methods offer a flexible framework for nonlinear transformations.

Purpose of the Study:

  • To model the nonlinear transformation in NLPCA using radially-symmetric kernel functions.
  • To develop a method for estimating the parameters of this variance-maximizing transformation.
  • To establish a cross-validation approach for model selection in NLPCA.

Main Methods:

  • Modeling the nonlinear transformation as a linear sum of kernel functions.
  • Minimizing a loss function that quantifies deviation from homogeneity.

Related Experiment Videos

  • Implementing an alternating least squares algorithm for parameter estimation.
  • Utilizing cross-validation for model selection.

    • Main Results:

    • The study demonstrates that NLPCA parameters can be found by minimizing a homogeneity-based loss function.
    • An effective alternating least squares algorithm is presented for this optimization.
    • The proposed method provides a basis for robust cross-validation in NLPCA.

    Conclusions:

    • Kernel-based modeling offers a powerful approach to NLPCA.
    • The developed algorithm and cross-validation routine facilitate accurate parameter estimation and model selection.
    • This work advances the application of nonlinear dimensionality reduction techniques.