We consider the problem of approximating a smooth function f from an Euclidean domain to a manifold M by scattered samples (f( i))i2I, where the data sites ( i)i2I are assumed to be locally close but can otherwise be far apart points scattered throughout the domain. We introduce a natural approximant based on combining the moving least square method and the Karcher mean. We prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart. The analysis also tells us that the use of Karcher’s mean (dependent on a Riemannian metric and the associated exponential map) is inessential and one can replace it by a more general notion of ‘center of mass’ based on a general retraction on the manifold. Consequently, we can substitute the Karcher mean by a more computationally ecient mean. We illustrate our work with numerical results which conrm our theoretical ndings.
['Philipp Grohs', 'Markus Sprecher', 'Thomas Yu']
Scattered manifold-valued data approximation