Manifold-valued Thin-plate Splines

In recent years a growing trend in machine learning is to go
beyond the standard setting of classification and regression and
to learn also in structured output spaces e.g. trees or graphs.

In this talk I will focus on another type of structured output
learning, namely where the output is manifold-valued. We present a
generalization of thin-plate splines for interpolation and
approximation of manifold-valued data. The cornerstone of our
theoretical framework is an energy functional for mappings between
two Riemannian manifolds which is independent of parameterization
and respects the geometry of both manifolds. If the manifolds are
Euclidean, the energy functional reduces to the classical
thin-plate spline energy. We show how the resulting optimization
problems can be solved efficiently in many cases. Our example
applications range from orientation interpolation and motion
planning in animation over geometric modeling tasks to color
interpolation.