Efficient Weingarten map and curvature estimation on manifolds
In this paper, we propose an efficient method to estimate the Weingarten map for point cloud data sampled from manifold embedded in Euclidean space. A statistical model is established to analyze the asymptotic property of the estimator. In particular, we show the convergence rate as the sample size...
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| Published in | Machine learning Vol. 110; no. 6; pp. 1319 - 1344 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.06.2021
Springer Nature B.V |
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| Abstract | In this paper, we propose an efficient method to estimate the Weingarten map for point cloud data sampled from manifold embedded in Euclidean space. A statistical model is established to analyze the asymptotic property of the estimator. In particular, we show the convergence rate as the sample size tends to infinity. We verify the convergence rate through simulated data and apply the estimated Weingarten map to curvature estimation and point cloud simplification to multiple real data sets. |
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| AbstractList | In this paper, we propose an efficient method to estimate the Weingarten map for point cloud data sampled from manifold embedded in Euclidean space. A statistical model is established to analyze the asymptotic property of the estimator. In particular, we show the convergence rate as the sample size tends to infinity. We verify the convergence rate through simulated data and apply the estimated Weingarten map to curvature estimation and point cloud simplification to multiple real data sets. |
| Author | Li, Didong Sun, Huafei Assadi, Amir H. Zhang, Shiqiang Cao, Yueqi |
| Author_xml | – sequence: 1 givenname: Yueqi surname: Cao fullname: Cao, Yueqi organization: School of Mathematics and Statistics, Beijing Institute of Technology – sequence: 2 givenname: Didong surname: Li fullname: Li, Didong organization: Department of Computer Science, Princeton University, Department of Biostatistics, University of California Los Angeles – sequence: 3 givenname: Huafei surname: Sun fullname: Sun, Huafei email: huafeisun@bit.edu.cn organization: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing Key Laboratory on Mathematical Characterization, Analysis and Applications of Complex Information – sequence: 4 givenname: Amir H. surname: Assadi fullname: Assadi, Amir H. organization: Department of Mathematics, University of Wisconsin Madison – sequence: 5 givenname: Shiqiang surname: Zhang fullname: Zhang, Shiqiang organization: School of Mathematics and Statistics, Beijing Institute of Technology |
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| Cites_doi | 10.1023/B:MACH.0000033120.25363.1e 10.1016/j.cviu.2006.09.007 10.1007/s00454-004-1146-y 10.1007/s00454-008-9053-2 10.1109/34.334391 10.1145/1138450.1138453 10.1007/s00454-002-2838-9 10.1090/conm/453/08802 10.1016/j.cagd.2005.06.010 10.1016/S0890-6955(01)00120-1 10.1109/TIP.2020.2969052 10.1016/j.cag.2017.07.015 10.1109/34.57680 10.1109/TPAMI.2002.1008395 10.1145/142920.134011 10.1016/S0097-8493(02)00121-8 10.1137/17M1112297 10.1016/j.neuroimage.2005.08.049 10.1137/15M1010506 10.1109/70.938384 10.1090/S0025-5718-98-00974-0 10.1007/s10711-005-9011-6 10.1137/140955483 10.1016/j.acha.2015.09.009 10.1002/cpa.21395 10.1016/S0167-8396(00)00006-6 10.1137/110845768 10.1109/TVCG.2010.261 10.1214/18-AOS1685 10.1016/j.neuroimage.2020.117295 10.1515/geofl-2018-0004 10.1111/cgf.12802 10.1007/978-3-642-40020-9_39 10.7551/mitpress/1120.003.0080 10.1007/11679363_37 10.7551/mitpress/7503.003.0021 10.1145/2090236.2090270 |
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| Keywords | Curvature estimation Weingarten map Point cloud data Unsupervised learning |
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| Snippet | In this paper, we propose an efficient method to estimate the Weingarten map for point cloud data sampled from manifold embedded in Euclidean space. A... |
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| SubjectTerms | Artificial Intelligence Asymptotic properties Computer Science Control Convergence Curvature Euclidean geometry Euclidean space Machine Learning Manifolds Mechatronics Natural Language Processing (NLP) Robotics Simulation and Modeling Statistical models |
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| Title | Efficient Weingarten map and curvature estimation on manifolds |
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