Journal ArticleDOI
Geodesic Regression and the Theory of Least Squares on Riemannian Manifolds
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TLDR
Specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age, which can be generally applied to data on any manifold.Abstract:
This paper develops the theory of geodesic regression and least-squares estimation on Riemannian manifolds. Geodesic regression is a method for finding the relationship between a real-valued independent variable and a manifold-valued dependent random variable, where this relationship is modeled as a geodesic curve on the manifold. Least-squares estimation is formulated intrinsically as a minimization of the sum-of-squared geodesic distances of the data to the estimated model. Geodesic regression is a direct generalization of linear regression to the manifold setting, and it provides a simple parameterization of the estimated relationship as an initial point and velocity, analogous to the intercept and slope. A nonparametric permutation test for determining the significance of the trend is also given. For the case of symmetric spaces, two main theoretical results are established. First, conditions for existence and uniqueness of the least-squares problem are provided. Second, a maximum likelihood criteria is developed for a suitable definition of Gaussian errors on the manifold. While the method can be generally applied to data on any manifold, specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age.read more
Citations
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Journal ArticleDOI
Bayesian Linear Size-and-Shape Regression with Applications to Face Data
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Proceedings ArticleDOI
Partial Matchings and Growth Mapped Evolutions in Shape Spaces
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Differentially Private Fréchet Mean on the Manifold of Symmetric Positive Definite (SPD) Matrices with log-Euclidean Metric
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Journal ArticleDOI
Kriging Riemannian Data via Random Domain Decompositions
TL;DR: Data taking value on a Riemannian manifold and observed over a complex spatial domain are becoming more frequent in applications, for example, in environmental sciences and in geoscience as discussed by the authors.
Journal ArticleDOI
Geodesic analysis in Kendall's shape space with epidemiological applications
Esfandiar Nava-Yazdani,Hans-Christian Hege,Timothy Sullivan,Timothy Sullivan,Christoph von Tycowicz +4 more
TL;DR: In this paper, the authors analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall's shape space using Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches.
References
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