Abstract
We present a probabilistic formulation for two closely related statistical models for Riemannian manifold data: geodesic regression and principal geodesic analysis. These models generalize linear regression and principal component analysis to the manifold setting. The foundation of the approach is the particular choice of a Riemannian normal distribution law as the likelihood model. Under this distributional assumption, we show that least-squares fitting of geodesic models is equivalent to maximum-likelihood estimation when the manifold is a homogeneous space. We also provide a method for maximum-likelihood estimation of the dispersion of the noise, as well as a novel method for Monte Carlo sampling from the Riemannian normal distribution. We demonstrate the inference procedures on synthetic sphere data, as well as in a shape analysis of the corpus callosum, represented in Kendall’s shape space.
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This work was supported in part by NSF CAREER Grant 1054057.
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Fletcher, P.T., Zhang, M. (2016). Probabilistic Geodesic Models for Regression and Dimensionality Reduction on Riemannian Manifolds. In: Turaga, P., Srivastava, A. (eds) Riemannian Computing in Computer Vision. Springer, Cham. https://doi.org/10.1007/978-3-319-22957-7_5
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DOI: https://doi.org/10.1007/978-3-319-22957-7_5
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