Before revision:  Thus, in case of extremely large ambient space dimension, we are able to avoid the curse of dimensionality without having to perform non-linear dimension reduction, which  introduces distortions to the manifold data. Using numerical experiments, we compare the presented method to state-of-the-art algorithms for regression over manifolds and show its potential.
After revision:  Thus, in case of extremely large ambient space dimension, we are able to avoid the curse of dimensionality without having to perform non-linear dimension reduction, which introduces distortions to the geometry of the data. Additionaly, we show numerical experiments that the proposed approach compares favorably to statistical approaches for regression over manifolds and show its potential.