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. Key words. harmonic map, biharmonic map, Eells energy, regularized empirical risk minimization, thin-plate spline AMS subject classi...

• Type: Journal
• Year: 2010
• Where: SIAMIS
• Authors: Florian Steinke, Matthias Hein, Bernhard Schölkopf
• Keywords: Empirical Risk Minimization, Manifolds, Riemannian Manifolds, SIAMIS 2010