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

arXiv: https://arxiv.org/abs/1801.04978 · HTML: https://mlsanity.com/paper/1801.04978/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jan 2018.

## Built on by (most cited first, 2 of 2)

- [Hessian Based Smoothing Splines for Manifold Learning](https://mlsanity.com/paper/2302.05025/) (2023)
- [Rolled Gaussian process models for curves on manifolds](https://mlsanity.com/paper/2503.21980/) (2025)

## Abstract (arXiv)

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 and unwrapping originally proposed in Jupp and Kent (1987) 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.