Approximation of Functions over Manifolds: A Moving Least-Squares Approach

We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. 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). The proposed algorithm is resistant to noise in both the domain and function values. Furthermore, the approximant is shown to be smooth and of approximation order of $\mathcal{O}(h^{m+1})$ for non-noisy data, where $h$ is the mesh size with respect to the manifold domain, and $m$ is the degree of a local polynomial approximation utilized in our algorithm. In addition, the proposed algorithm is linear in time with respect to the ambient-space's dimension. Thus, in case of extremely large ambient space dimension, we are able to avoid the curse of dimensionality without having to perform non-linear dimension reduction, which introduces distortions to the manifold data. Using numerical experiments, we compare the presented method to state-of-the-art algorithms for regression over manifolds and show its potential.