Manifold-Valued Function Approximation from Multiple Tangent Spaces

Abstract. 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 Fréchet mean. The effectiveness of this approach is illustrated with numerical experiments on model problems from parametric model order reduction. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and Data Available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://gitlab.kuleuven.be/numa/public/mtsm as well as in the supplemental material ( mtsm-main.zip [29.8KB]). [Formula: see text]