When Locally Linear Embedding Hits Boundary

Fuente: arXiv
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Auteurs principaux: Wu, Hau-tieng, Wu, Nan
Format: Preprint
Publié: 2018
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author Wu, Hau-tieng
Wu, Nan
author_facet Wu, Hau-tieng
Wu, Nan
contents 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.
format Preprint
id arxiv_https___arxiv_org_abs_1811_04423
institution arXiv
publishDate 2018
record_format arxiv
spellingShingle When Locally Linear Embedding Hits Boundary
Wu, Hau-tieng
Wu, Nan
Statistics Theory
62-07
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.
title When Locally Linear Embedding Hits Boundary
topic Statistics Theory
62-07
url https://arxiv.org/abs/1811.04423