Think globally, fit locally under the manifold setup: Asymptotic analysis of locally linear embedding

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.