Approximation of functions on manifolds in high dimension from noisy scattered data

In this paper, we consider the fundamental problem of approximation of functions on a low-dimensional manifold embedded in a high-dimensional space. Classical approximation methods, developed for the low-dimensional case, are challenged by the high-dimensional data, and the presence of noise. Here, we introduce a new approximation method that is parametrization free, can handle noise and outliers in both the scattered data and function values and does not require any assumptions on the scattered data geometry. Given a noisy point-cloud situated near a low dimensional manifold and the corresponding noisy function values, the proposed solution finds a noise-free, quasi-uniform manifold reconstruction as well as the denoised function values at these points. Next, this data is used to approximate the function at new points near the manifold. We prove that in the case of noise-free samples the approximation order is O(h²), where h depends on the local density of the dataset (i.e., the fill distance), and the function variation.