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