@article{f2a190b882a94c4fb582c862ad003832,
title = "A level set based variational principal flow method for nonparametric dimension reduction on Riemannian manifolds",
abstract = "We propose a variational formulation for dimension reduction on Riemannian manifolds. The algorithm is developed based on the level set method together with a recently developed principal flow algorithm. The original principal flow algorithm is a Lagrangian technique which extends the principal component analysis (PCA) to dimension reduction on Riemannian manifolds. We propose to incorporate the level set method to obtain a fully implicit formulation so that the overall algorithm can naturally handle various topological changes in the curve evolution. The variational formulation consists of two terms which try to balance the contributions from both the dataset itself and the principal direction by the PCA. We will demonstrate that the method is insensitive to the initial guess and is robust enough for noisy data.",
keywords = "Dimensional reduction, Implicit surfaces, Level set method, Partial differential equations, PCA",
author = "Hao Liu and Zhigang Yao and Shingyu Leung and Chan, {Tony F.}",
note = "Funding Information: ∗Submitted to the journal{\textquoteright}s Methods and Algorithms for Scientific Computing section April 26, 2016; accepted for publication (in revised form) June 13, 2017; published electronically August 24, 2017. http://www.siam.org/journals/sisc/39-4/M107236.html Funding: The second author{\textquoteright}s work was supported by MOE Tier 1 (R-155-000-154-133) and MOE Tier 2 (R-155-000-184-112). The third author{\textquoteright}s work was supported in part by the Hong Kong RGC grants 16303114 and 16309316. †Department of Mathematics, the Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong (
[email protected],
[email protected]). ‡Department of Statistics and Applied Probability, National University of Singapore, 21 Lower Kent Ridge Road, Singapore 117546 (
[email protected]). §Office of the President, the Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong (
[email protected]). Publisher Copyright: {\textcopyright} 2017 SIAM.",
year = "2017",
month = aug,
doi = "10.1137/16M107236X",
language = "English",
volume = "39",
pages = "A1616--A1646",
journal = "SIAM Journal on Scientific Computing",
issn = "1064-8275",
publisher = "Society for Industrial and Applied Mathematics (SIAM)",
number = "4",
}