Title: Approximation of Functions over Manifolds: A Moving Least-Squares Approach,
Abstract: 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.