TY - JOUR
T1 - Approximation of functions on manifolds in high dimension from noisy scattered data
AU - Faigenbaum-Golovin, Shira
AU - Levin, David
N1 - Publisher Copyright:
© 2022 Universidad de Jaén
PY - 2022
Y1 - 2022
N2 - In this paper, we consider the fundamental problem of approximation of functions on a low-dimensional manifold embedded in a high-dimensional space. Classical approximation methods, developed for the low-dimensional case, are challenged by the high-dimensional data, and the presence of noise. Here, we introduce a new approximation method that is parametrization free, can handle noise and outliers in both the scattered data and function values and does not require any assumptions on the scattered data geometry. Given a noisy point-cloud situated near a low dimensional manifold and the corresponding noisy function values, the proposed solution finds a noise-free, quasi-uniform manifold reconstruction as well as the denoised function values at these points. Next, this data is used to approximate the function at new points near the manifold. We prove that in the case of noise-free samples the approximation order is O(h2), where h depends on the local density of the dataset (i.e., the fill distance), and the function variation.
AB - In this paper, we consider the fundamental problem of approximation of functions on a low-dimensional manifold embedded in a high-dimensional space. Classical approximation methods, developed for the low-dimensional case, are challenged by the high-dimensional data, and the presence of noise. Here, we introduce a new approximation method that is parametrization free, can handle noise and outliers in both the scattered data and function values and does not require any assumptions on the scattered data geometry. Given a noisy point-cloud situated near a low dimensional manifold and the corresponding noisy function values, the proposed solution finds a noise-free, quasi-uniform manifold reconstruction as well as the denoised function values at these points. Next, this data is used to approximate the function at new points near the manifold. We prove that in the case of noise-free samples the approximation order is O(h2), where h depends on the local density of the dataset (i.e., the fill distance), and the function variation.
KW - approximation of functions
KW - dimensional reduction
KW - high dimensions
KW - manifold learning
KW - noisy data
KW - scattered data
UR - http://www.scopus.com/inward/record.url?scp=85156255067&partnerID=8YFLogxK
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AN - SCOPUS:85156255067
SN - 1889-3066
VL - 13
SP - 43
EP - 73
JO - Jaen Journal on Approximation
JF - Jaen Journal on Approximation
ER -