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Nonparametric Regression between General Riemannian Manifolds




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.

Author(s): Steinke, F. and Hein, M. and Schölkopf, B.
Journal: SIAM Journal on Imaging Sciences
Volume: 3
Number (issue): 3
Pages: 527-563
Year: 2010
Month: September
Day: 0

Department(s): Empirical Inference
Bibtex Type: Article (article)

Digital: 0
DOI: 10.1137/080744189
Language: en
Organization: Max-Planck-Gesellschaft
School: Biologische Kybernetik

Links: Web


  title = {Nonparametric Regression between General Riemannian Manifolds},
  author = {Steinke, F. and Hein, M. and Sch{\"o}lkopf, B.},
  journal = {SIAM Journal on Imaging Sciences},
  volume = {3},
  number = {3},
  pages = {527-563},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  month = sep,
  year = {2010},
  month_numeric = {9}