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

Hau-Tieng Wu, Nan Wu
2018 Annals of Statistics  
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 non-uniform 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
more » ... applied nonlinear algorithms, particularly diffusion map, is provided, and its relationship with locally linear regression is also discussed. MSC 2010 subject classifications: Primary 60K35, 60K35; secondary 60K35
doi:10.1214/17-aos1676 fatcat:lsjcfx7jdfeahm6kl4kmumguda