Think globally, fit locally under the Manifold Setup: Asymptotic Analysis of Locally Linear Embedding

2017/03/12 by Hau‐Tieng Wu, Nan Wu, Wu, Hau-Tieng +1 · 1 citation
Computer Science · Mathematics · #62-07 #Bayesian Methods and Mixture Models #FOS: Mathematics #Face and Expression Recognition #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1703.04058

openalex publication_date 2017/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Spectral Convergence Rate of Graph Laplacian
• Connection graph Laplacian methods can be made robust to noise

Cited by

• Statistical Inference for Manifold Similarity and Alignability across Noisy High-Dimensional Datasets

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