When Locally Linear Embedding Hits Boundary

Abstract

Based on the Riemannian manifold model, we study the asymptotic behavior of a widely applied unsupervised learning algorithm, locally linear embedding (LLE), when the point cloud is sampled from a compact, smooth manifold with boundary. We show several peculiar behaviors of LLE near the boundary that are different from those diffusion-based algorithms. In particular, we show that LLE pointwisely converges to a mixed-type differential operator with degeneracy and we calculate the convergence rate. The impact of the hyperbolic part of the operator is discussed and we propose a clipped LLE algorithm which is a potential approach to recover the Dirichlet Laplace-Beltrami operator.

70 Pages, 11 Figures

References in corpus (3)

• Consistency of spectral clustering
• Learning gradients on manifolds
• Recovering Hidden Components in Multimodal Data with Composite Diffusion Operators

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