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In this paper, we consider the fundamental problem of approximation offunctions 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 thecurse of dimensionality, as well as to the presence of noise, the classicalapproximation methods applicable in low dimensions are less effective in thehigh-dimensional case. We propose a new approximation method that leverages theadvantages of the Manifold Locally Optimal Projection (MLOP) method (introducedby Faigenbaum-Golovin and Levin in 2020) and the strengths of the method ofRadial Basis Functions (RBF). The method is parametrization free, requires noknowledge regarding the manifold's intrinsic dimension, can handle noise andoutliers in both the function values and in the location of the data, and isapplied directly in the high dimensions. We show that the complexity of themethod is linear in the dimension of the manifold and squared-logarithmic inthe dimension of the codomain of the function. Subsequently, we demonstrate theeffectiveness of our approach by considering different manifold topologies andshow the robustness of the method to various noise levels.

Approximation of Functions on Manifolds in High Dimension from Noisy Scattered Data