Locally Linear Embedding on Boundary of Riemannian Manifolds
By Hau-Tieng Wu and Nan Wu; Published in 2023, Volume 24, Issue 69, Pages 1-80.
Abstract
This study investigates the behavior of the locally linear embedding (LLE) algorithm, a commonly used unsupervised learning technique, when applied to point clouds sampled from compact, smooth manifolds with boundaries. The analysis is based on the Riemannian manifold model. We identify several distinct characteristics of LLE near the boundary, which differ from diffusion-based algorithms. Specifically, we demonstrate that LLE converges pointwisely to a mixed-type differential operator with degeneracy, and we determine the convergence rate. Additionally, we discuss the impact of the hyperbolic part of the operator and propose a clipped LLE algorithm as a potential approach for recovering the Dirichlet Laplace-Beltrami operator.
[Abstract]
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