Manifold-valued function approximation from multiple tangent spaces

Approximating a manifold-valued function from samples of input-output pairs consists of modeling the relationship between an input from a vector space and an output on a Riemannian manifold. We propose a function approximation method that leverages and unifies two prior techniques: (i) approximating a pullback to the tangent space, and (ii) the Riemannian moving least squares method. The core idea of the new scheme is to combine pullbacks to multiple tangent spaces with a weighted Fr\'echet mean. The effectiveness of this approach is illustrated with numerical experiments on model problems from parametric model order reduction.