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Gradient-Based Kernel Dimension Reduction for Regression

Author

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  • Kenji Fukumizu
  • Chenlei Leng

Abstract

This article proposes a novel approach to linear dimension reduction for regression using nonparametric estimation with positive-definite kernels or reproducing kernel Hilbert spaces (RKHSs). The purpose of the dimension reduction is to find such directions in the explanatory variables that explain the response sufficiently: this is called sufficient dimension reduction . The proposed method is based on an estimator for the gradient of the regression function considered for the feature vectors mapped into RKHSs. It is proved that the method is able to estimate the directions that achieve sufficient dimension reduction. In comparison with other existing methods, the proposed one has wide applicability without strong assumptions on the distributions or the type of variables, and needs only eigendecomposition for estimating the projection matrix. The theoretical analysis shows that the estimator is consistent with certain rate under some conditions. The experimental results demonstrate that the proposed method successfully finds effective directions with efficient computation even for high-dimensional explanatory variables.

Suggested Citation

  • Kenji Fukumizu & Chenlei Leng, 2014. "Gradient-Based Kernel Dimension Reduction for Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(505), pages 359-370, March.
  • Handle: RePEc:taf:jnlasa:v:109:y:2014:i:505:p:359-370
    DOI: 10.1080/01621459.2013.838167
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    Cited by:

    1. Mehni, Moien Barkhori & Mehni, Mohammad Barkhori, 2023. "Reliability analysis with cross-entropy based adaptive Markov chain importance sampling and control variates," Reliability Engineering and System Safety, Elsevier, vol. 231(C).
    2. Alam, Md. Ashad & Calhoun, Vince D. & Wang, Yu-Ping, 2018. "Identifying outliers using multiple kernel canonical correlation analysis with application to imaging genetics," Computational Statistics & Data Analysis, Elsevier, vol. 125(C), pages 70-85.
    3. He, Xin & Mao, Xiaojun & Wang, Zhonglei, 2024. "Nonparametric augmented probability weighting with sparsity," Computational Statistics & Data Analysis, Elsevier, vol. 191(C).
    4. Zuniga, M. Munoz & Murangira, A. & Perdrizet, T., 2021. "Structural reliability assessment through surrogate based importance sampling with dimension reduction," Reliability Engineering and System Safety, Elsevier, vol. 207(C).
    5. Ming-Yueh Huang & Kwun Chuen Gary Chan, 2017. "Joint sufficient dimension reduction and estimation of conditional and average treatment effects," Biometrika, Biometrika Trust, vol. 104(3), pages 583-596.
    6. Strobl Eric V. & Visweswaran Shyam, 2016. "Markov Boundary Discovery with Ridge Regularized Linear Models," Journal of Causal Inference, De Gruyter, vol. 4(1), pages 31-48, March.
    7. Sadikoglu, Serhan, 2019. "Essays in econometric theory," Other publications TiSEM 99d83644-f9dc-49e3-a4e1-5, Tilburg University, School of Economics and Management.
    8. Cizek, Pavel & Sadikoglu, Serhan, 2022. "Nonseparable Panel Models with Index Structure and Correlated Random Effects," Other publications TiSEM 7899deb9-0eda-47e6-a3b8-2, Tilburg University, School of Economics and Management.

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