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Local Linear Regression on Manifolds and Its Geometric Interpretation

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  • Ming-yen Cheng
  • Hau-tieng Wu

Abstract

High-dimensional data analysis has been an active area, and the main focus areas have been variable selection and dimension reduction. In practice, it occurs often that the variables are located on an unknown, lower-dimensional nonlinear manifold. Under this manifold assumption, one purpose of this article is regression and gradient estimation on the manifold, and another is developing a new tool for manifold learning. As regards the first aim, we suggest directly reducing the dimensionality to the intrinsic dimension d of the manifold, and performing the popular local linear regression (LLR) on a tangent plane estimate. An immediate consequence is a dramatic reduction in the computational time when the ambient space dimension p >> d . We provide rigorous theoretical justification of the convergence of the proposed regression and gradient estimators by carefully analyzing the curvature, boundary, and nonuniform sampling effects. We propose a bandwidth selector that can handle heteroscedastic errors. With reference to the second aim, we analyze carefully the asymptotic behavior of our regression estimator both in the interior and near the boundary of the manifold, and make explicit its relationship with manifold learning, in particular estimating the Laplace--Beltrami operator of the manifold. In this context, we also make clear that it is important to use a smaller bandwidth in the tangent plane estimation than in the LLR. A simulation study and applications to the Isomap face data and a clinically computed tomography scan dataset are used to illustrate the computational speed and estimation accuracy of our methods. Supplementary materials for this article are available online.

Suggested Citation

  • Ming-yen Cheng & Hau-tieng Wu, 2013. "Local Linear Regression on Manifolds and Its Geometric Interpretation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(504), pages 1421-1434, December.
  • Handle: RePEc:taf:jnlasa:v:108:y:2013:i:504:p:1421-1434
    DOI: 10.1080/01621459.2013.827984
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    Citations

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    Cited by:

    1. Gouriéroux, Christian & Monfort, Alain & Zakoian, Jean-Michel, 2017. "Pseudo-Maximum Likelihood and Lie Groups of Linear Transformations," MPRA Paper 79623, University Library of Munich, Germany.
    2. Lizhen Lin & Brian St. Thomas & Hongtu Zhu & David B. Dunson, 2017. "Extrinsic Local Regression on Manifold-Valued Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1261-1273, July.
    3. Cui, Xia & Lu, Ying & Peng, Heng, 2017. "Estimation of partially linear regression models under the partial consistency property," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 103-121.
    4. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.
    5. Yiyi Huo & Yingying Fan & Fang Han, 2023. "On the adaptation of causal forests to manifold data," Papers 2311.16486, arXiv.org, revised Dec 2023.
    6. David B. Dunson & Hau‐Tieng Wu & Nan Wu, 2022. "Graph based Gaussian processes on restricted domains," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 414-439, April.
    7. Eduardo GarcÍa-Portugués & Ingrid Van Keilegom & Rosa M. Crujeiras and & Wenceslao González-Manteiga, 2016. "Testing parametric models in linear-directional regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(4), pages 1178-1191, December.

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