A1 Journal article (refereed)
Approximation of functions over manifolds : A Moving Least-Squares approach (2021)
Sober, B., Aizenbud, Y., & Levin, D. (2021). Approximation of functions over manifolds : A Moving Least-Squares approach. Journal of Computational and Applied Mathematics, 383, Article 113140. https://doi.org/10.1016/j.cam.2020.113140
JYU authors or editors
Publication details
All authors or editors: Sober, Barak; Aizenbud, Yariv; Levin, David
Journal or series: Journal of Computational and Applied Mathematics
ISSN: 0377-0427
eISSN: 1879-1778
Publication year: 2021
Volume: 383
Article number: 113140
Publisher: Elsevier BV
Publication country: Netherlands
Publication language: English
DOI: https://doi.org/10.1016/j.cam.2020.113140
Publication open access: Not open
Publication channel open access:
Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/77962
Web address of parallel published publication (pre-print): https://arxiv.org/abs/1711.00765
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 knowledge about the local geometry of the manifold or its local parameterizations. We do require, however, knowledge regarding the manifold's intrinsic dimension d. We use the Manifold Moving Least-Squares approach of Sober and Levin (2019) 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(hm+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. Additionally, we show numerically that our approach compares favorably to some well-known approaches for regression over manifolds.
Keywords: numerical analysis; approximation; functions (mathematical methods); manifolds (mathematics)
Contributing organizations
Ministry reporting: Yes
Reporting Year: 2021
JUFO rating: 2