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December 2018 Think globally, fit locally under the manifold setup: Asymptotic analysis of locally linear embedding
Hau-Tieng Wu, Nan Wu
Ann. Statist. 46(6B): 3805-3837 (December 2018). DOI: 10.1214/17-AOS1676

Abstract

Since its introduction in 2000, Locally Linear Embedding (LLE) has been widely applied in data science. We provide an asymptotical analysis of LLE under the manifold setup. We show that for a general manifold, asymptotically we may not obtain the Laplace–Beltrami operator, and the result may depend on nonuniform sampling unless a correct regularization is chosen. We also derive the corresponding kernel function, which indicates that LLE is not a Markov process. A comparison with other commonly applied nonlinear algorithms, particularly a diffusion map, is provided and its relationship with locally linear regression is also discussed.

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Hau-Tieng Wu. Nan Wu. "Think globally, fit locally under the manifold setup: Asymptotic analysis of locally linear embedding." Ann. Statist. 46 (6B) 3805 - 3837, December 2018. https://doi.org/10.1214/17-AOS1676

Information

Received: 1 March 2017; Revised: 1 December 2017; Published: December 2018
First available in Project Euclid: 11 September 2018

zbMATH: 1405.62058
MathSciNet: MR3852669
Digital Object Identifier: 10.1214/17-AOS1676

Subjects:
Primary: 60K35

Keywords: diffusion maps , Dimension reduction , Locally linear embedding , locally linear regression , measurement error

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 6B • December 2018
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