Approximation of functions over manifolds: A Moving Least-Squares approach

We present an algorithm for approximating a function defined over $d$-dimensional manifold utilizing only noisy values at locations sampled from the with noise. To produce approximation we do not require any knowledge regarding other than its dimension $d$. use Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct atlas charts is built on-top those charts. The resulting approximant shown be neighborhood manifold, originally manifold. In words, given new point, located near can evaluated directly on that point. prove our construction yields smooth function, in case noiseless samples order $\mathcal{O}(h^{m+1})$, where $h$ local density sample parameter (i.e., fill distance) $m$ degree polynomial approximation, used algorithm. addition, proposed has linear time complexity respect ambient-space's dimension. Thus, are able avoid computational complexity, commonly encountered high dimensional approximations, without having perform non-linear reduction, which inevitably introduces distortions geometry data. Additionaly, show numerical experiments compares favorably statistical approaches regression manifolds potential.