Smoothing splines on Riemannian manifolds, with applications to 3D shape space

Kim, Kwang Rae; Dryden, Ian L.; Le, Huiling; Severn, Katie E.

Abstract

© 2020 The Authors. Journal of the Royal Statistical Society: Series B (Statistical Methodology) published by John Wiley & Sons Ltd on behalf of Royal Statistical Society There has been increasing interest in statistical analysis of data lying in manifolds. This paper generalizes a smoothing spline fitting method to Riemannian manifold data based on the technique of unrolling, unwrapping and wrapping originally proposed by Jupp and Kent for spherical data. In particular, we develop such a fitting procedure for shapes of configurations in general m-dimensional Euclidean space, extending our previous work for two-dimensional shapes. We show that parallel transport along a geodesic on Kendall shape space is linked to the solution of a homogeneous first-order differential equation, some of whose coefficients are implicitly defined functions. This finding enables us to approximate the procedure of unrolling and unwrapping by simultaneously solving such equations numerically, and so to find numerical solutions for smoothing splines fitted to higher dimensional shape data. This fitting method is applied to the analysis of some dynamic 3D peptide data.

Citation

Kim, K. R., Dryden, I. L., Le, H., & Severn, K. E. (2021). Smoothing splines on Riemannian manifolds, with applications to 3D shape space. Journal of the Royal Statistical Society: Series B, 83(1), 108-132. https://doi.org/10.1111/rssb.12402

• Journal Article Type: Article
• Acceptance Date: Oct 15, 2020
• Online Publication Date: Dec 2, 2020
• Publication Date: 2021-02
• Deposit Date: Oct 16, 2020
• Publicly Available Date: Dec 3, 2021
• Journal: Journal of the Royal Statistical Society: Series B (Statistical Methodology)
• Print ISSN: 1369-7412
• Electronic ISSN: 1467-9868
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 83
• Issue: 1
• Pages: 108-132
• DOI: https://doi.org/10.1111/rssb.12402
• Keywords: cubic spline; geodesic; non-parametric regression; linear spline; parallel transport; peptide; tangent space; unrolling; unwrapping; wrapping.
• Public URL: https://nottingham-repository.worktribe.com/output/2458450
• Publisher URL: https://rss.onlinelibrary.wiley.com/doi/full/10.1111/rssb.12402