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 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.
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. We use the Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated directly on that point. We prove that our construction yields a smooth function, and in case of noiseless samples the approximation order is O(h^{m+1}) , where h is a local density of sample parameter (i.e., the fill distance) and m is the degree of a local polynomial approximation , used in our algorithm. In addition, the proposed algorithm has linear time complexity with respect to the ambient-space's dimension. Thus, we are able to avoid the computational complexity, commonly encountered in high dimensional approximations, without having to perform non-linear dimension reduction, which inevitably introduces distortions to the geometry of the data. Additionaly, we show numerical experiments that the proposed approach compares favorably to statistical approaches for regression over manifolds and show its potential.
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The approximation scheme is based upon
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(MMLS) . The proposed algorithm is resistant to noise in both the domain and function values. Furthermore, the
approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting
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smooth and of approximation order of
a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated directly on that point. We prove that our construction yields a smooth function, and in case of noiseless samples the approximation order is
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for non-noisy data
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the mesh size with respect to the manifold domain,
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utilized
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is linear in time
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in case of extremely large ambient space dimension,
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curse of dimensionality
computational complexity, commonly encountered in high dimensional approximations,
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inevitably
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manifold data. Using numerical experiments, we compare the presented method to state-of-the-art algorithms
geometry of the data. Additionaly, we show numerical experiments that the proposed approach compares favorably to statistical approaches