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Extrinsic Local Regression on Manifold-Valued Data

Author

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  • Lizhen Lin
  • Brian St. Thomas
  • Hongtu Zhu
  • David B. Dunson

Abstract

We propose an extrinsic regression framework for modeling data with manifold valued responses and Euclidean predictors. Regression with manifold responses has wide applications in shape analysis, neuroscience, medical imaging, and many other areas. Our approach embeds the manifold where the responses lie onto a higher dimensional Euclidean space, obtains a local regression estimate in that space, and then projects this estimate back onto the image of the manifold. Outside the regression setting both intrinsic and extrinsic approaches have been proposed for modeling iid manifold-valued data. However, to our knowledge our work is the first to take an extrinsic approach to the regression problem. The proposed extrinsic regression framework is general, computationally efficient, and theoretically appealing. Asymptotic distributions and convergence rates of the extrinsic regression estimates are derived and a large class of examples is considered indicating the wide applicability of our approach. Supplementary materials for this article are available online.

Suggested Citation

  • 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.
  • Handle: RePEc:taf:jnlasa:v:112:y:2017:i:519:p:1261-1273
    DOI: 10.1080/01621459.2016.1208615
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    References listed on IDEAS

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    1. Pelletier, Bruno, 2005. "Kernel density estimation on Riemannian manifolds," Statistics & Probability Letters, Elsevier, vol. 73(3), pages 297-304, July.
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    4. Chao Huang & Martin Styner & Hongtu Zhu, 2015. "Clustering High-Dimensional Landmark-Based Two-Dimensional Shape Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(511), pages 946-961, September.
    5. 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.
    6. Marco Di Marzio & Agnese Panzera & Charles C. Taylor, 2014. "Nonparametric Regression for Spherical Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(506), pages 748-763, June.
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    Cited by:

    1. Kwang‐Rae Kim & Ian L. Dryden & Huiling Le & Katie E. Severn, 2021. "Smoothing splines on Riemannian manifolds, with applications to 3D shape space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 108-132, February.
    2. Rabi Bhattacharya & Rachel Oliver, 2019. "Nonparametric Analysis of Non-Euclidean Data on Shapes and Images," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 1-36, February.
    3. Arthur Pewsey & Eduardo García-Portugués, 2021. "Rejoinder on: 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 76-82, March.
    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. Xiongtao Dai & Zhenhua Lin & Hans‐Georg Müller, 2021. "Modeling sparse longitudinal data on Riemannian manifolds," Biometrics, The International Biometric Society, vol. 77(4), pages 1328-1341, December.
    6. S. Barahona & P. Centella & X. Gual-Arnau & M. V. Ibáñez & A. Simó, 2020. "Supervised classification of geometrical objects by integrating currents and functional data analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(3), pages 637-660, September.
    7. Stephan F. Huckemann, 2021. "Comments on: 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 71-75, March.
    8. Vic Patrangenaru & Yifang Deng, 2021. "Extrinsic Regression and Anti-Regression on Projective Shape Manifolds," Methodology and Computing in Applied Probability, Springer, vol. 23(2), pages 629-646, June.
    9. Fraiman, Ricardo & Gamboa, Fabrice & Moreno, Leonardo, 2019. "Connecting pairwise geodesic spheres by depth: DCOPS," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 81-94.

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