# Nonlinear Regression on Manifolds for Shape Analysis using Intrinsic B'ezier Splines

arXiv: https://arxiv.org/abs/2007.05275 · HTML: https://mlsanity.com/paper/2007.05275/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jul 2020.

## Built on by (most cited first, 5 of 5)

- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Sasaki Metric for Spline Models of Manifold-Valued Trajectories](https://mlsanity.com/paper/2303.17299/) (2023)
- [Geodesic B-Score for Improved Assessment of Knee Osteoarthritis](https://mlsanity.com/paper/2104.01107/) (2021)
- [Intrinsic shape analysis in archaeology: A case study on ancient sundials](https://mlsanity.com/paper/2305.18960/) (2023)
- [Bi-invariant Geodesic Regression with Data from the Osteoarthritis Initiative](https://mlsanity.com/paper/2502.11826/) (2025)

## Abstract (arXiv)

Intrinsic and parametric regression models are of high interest for the statistical analysis of manifold-valued data such as images and shapes. The standard linear ansatz has been generalized to geodesic regression on manifolds making it possible to analyze dependencies of random variables that spread along generalized straight lines. Nevertheless, in some scenarios, the evolution of the data cannot be modeled adequately by a geodesic. We present a framework for nonlinear regression on manifolds by considering Riemannian splines, whose segments are B'ezier curves, as trajectories. Unlike variational formulations that require time-discretization, we take a constructive approach that provides efficient and exact evaluation by virtue of the generalized de Casteljau algorithm. We validate our method in experiments on the reconstruction of periodic motion of the mitral valve as well as the analysis of femoral shape changes during the course of osteoarthritis, endorsing B'ezier spline regression as an effective and flexible tool for manifold-valued regression.