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Nonparametric Regression between General Riemannian Manifolds

MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons84235

Steinke,  F
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons83958

Hein,  M
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons84193

Schölkopf,  B
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Steinke, F., Hein, M., & Schölkopf, B. (2010). Nonparametric Regression between General Riemannian Manifolds. SIAM Journal on Imaging Sciences, 3(3), 527-563. doi:10.1137/080744189.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-BE54-5
Abstract
We study nonparametric regression between Riemannian manifolds based on regularized empirical risk minimization. Regularization functionals for mappings between manifolds should respect the geometry of input and output manifold and be independent of the chosen parametrization of the manifolds. We define and analyze the three most simple regularization functionals with these properties and present a rather general scheme for solving the resulting optimization problem. As application examples we discuss interpolation on the sphere, fingerprint processing, and correspondence computations between three-dimensional surfaces. We conclude with characterizing interesting and sometimes counterintuitive implications and new open problems that are specific to learning between Riemannian manifolds and are not encountered in multivariate regression in Euclidean space.