Approximation of Functions Over Manifolds by Moving Least Squares

We approximate a function defined over a d-dimensional manifold M ⊂Rn utilizing only noisy function values at noisy locations on the manifold. To produce the approximation, we do not require any knowledge regarding the manifold other than its dimension d. The approximation scheme is based upon the Manifold Moving Least-Squares (MMLS) and is therefore resistant to noise in the domain M as well. Furthermore, the approximant is shown to be smooth and of approximation order of O(hm+1) for non-noisy data, where h is the mesh size w.r.t M, and m is the degree of the local polynomial approximation. In addition, the proposed algorithm is linear in time with respect to the ambient space dimension n, making it useful for cases where d<