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 -