  In this paper, we consider the fundamental problem of approximation of
functions on a low-dimensional manifold embedded in a high-dimensional space,
with noise affecting both in the data and values of the functions. Due to the
curse of dimensionality, as well as to the presence of noise, the classical
approximation methods applicable in low dimensions are less effective in the
high-dimensional case. We propose a new approximation method that leverages the
advantages of the Manifold Locally Optimal Projection (MLOP) method (introduced
by Faigenbaum-Golovin and Levin in 2020) and the strengths of the method of
Radial Basis Functions (RBF). The method is parametrization free, requires no
knowledge regarding the manifold's intrinsic dimension, can handle noise and
outliers in both the function values and in the location of the data, and is
applied directly in the high dimensions. We show that the complexity of the
method is linear in the dimension of the manifold and squared-logarithmic in
the dimension of the codomain of the function. Subsequently, we demonstrate the
effectiveness of our approach by considering different manifold topologies and
show the robustness of the method to various noise levels.
