Learning on manifolds without manifold learning

Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. In contrast to the prevalent paradigm of solving this problem by minimizing a loss functional, we have given a direct one-shot construction together with optimal error bounds under the manifold assumption; i.e., one assumes that the data is sampled from an unknown sub-manifold of a high dimensional Euclidean space. A great deal of research deals with obtaining information about this manifold, such as the eigendecomposition of the Laplace-Beltrami operator or coordinate charts, and using this information for function approximation. This two step approach implies some extra errors in the approximation stemming from basic quantities of the data in addition to the errors inherent in function approximation. In Neural Networks, 132:253268, 2020, we have proposed a one-shot direct method to achieve function approximation without requiring the extraction of any information about the manifold other than its dimension. However, one cannot pin down the class of approximants used in that paper. In this paper, we view the unknown manifold as a sub-manifold of an ambient hypersphere and study the question of constructing a one-shot approximation using the spherical polynomials based on the hypersphere. Our approach does not require preprocessing of the data to obtain information about the manifold other than its dimension. We give optimal rates of approximation for relatively "rough" functions.