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Some aspects of nonlinear dimensionality reduction

Author

Listed:
  • Liwen Wang

    (Ministry of Education, School of Science, Beijing University of Posts and Telecommunications)

  • Yongda Wang

    (University of Chinese Academy of Sciences
    Chinese Academy of Sciences)

  • Shifeng Xiong

    (University of Chinese Academy of Sciences
    Chinese Academy of Sciences)

  • Jiankui Yang

    (Ministry of Education, School of Science, Beijing University of Posts and Telecommunications)

Abstract

In this paper we discuss nonlinear dimensionality reduction within the framework of principal curves. We formulate dimensionality reduction as problems of estimating principal subspaces for both noiseless and noisy cases, and propose the corresponding iterative algorithms that modify existing principal curve algorithms. An R squared criterion is introduced to estimate the dimension of the principal subspace. In addition, we present new regression and density estimation strategies based on our dimensionality reduction algorithms. Theoretical analyses and numerical experiments show the effectiveness of the proposed methods.

Suggested Citation

  • Liwen Wang & Yongda Wang & Shifeng Xiong & Jiankui Yang, 2025. "Some aspects of nonlinear dimensionality reduction," Computational Statistics, Springer, vol. 40(2), pages 883-906, February.
    • Handle: RePEc:spr:compst:v:40:y:2025:i:2:d:10.1007_s00180-024-01514-0
    DOI: 10.1007/s00180-024-01514-0
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    References listed on IDEAS

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    1. Delicado, Pedro, 2001. "Another Look at Principal Curves and Surfaces," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 84-116, April.
    2. Chen Yue & Vadim Zipunnikov & Pierre-Louis Bazin & Dzung Pham & Daniel Reich & Ciprian Crainiceanu & Brian Caffo, 2016. "Parameterization of White Matter Manifold-Like Structures Using Principal Surfaces," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(515), pages 1050-1060, July.
    3. Mühlenstädt, Thomas & Kuhnt, Sonja, 2011. "Kernel interpolation," Computational Statistics & Data Analysis, Elsevier, vol. 55(11), pages 2962-2974, November.
    4. Mu, Weiyan & Xiong, Shifeng, 2018. "A class of space-filling designs and their projection properties," Statistics & Probability Letters, Elsevier, vol. 141(C), pages 129-134.
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