Geodesic Regression on Riemannian Manifolds

Abstract : This paper introduces a regression method for modeling the relationship between a manifold-valued random variable and a real-valued independent parameter. The principle is to fit a geodesic curve, parameterized by the independent parameter, that best fits the data. Error in the model is evaluated as the sum-of-squared geodesic distances from the model to the data, and this provides an intrinsic least squares criterion. Geodesic regression is, in some sense, the simplest parametric model that one could choose, and it provides a direct generalization of linear regression to the manifold setting. A hypothesis test for determining the significance of the estimated trend is also developed. 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.
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Pennec, Xavier and Joshi, Sarang and Nielsen, Mads. Proceedings of the Third International Workshop on Mathematical Foundations of Computational Anatomy - Geometrical and Statistical Methods for Modelling Biological Shape Variability, Sep 2011, Toronto, Canada. pp.75-86, 2011
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Thomas Fletcher. Geodesic Regression on Riemannian Manifolds. Pennec, Xavier and Joshi, Sarang and Nielsen, Mads. Proceedings of the Third International Workshop on Mathematical Foundations of Computational Anatomy - Geometrical and Statistical Methods for Modelling Biological Shape Variability, Sep 2011, Toronto, Canada. pp.75-86, 2011. 〈inria-00623920〉

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