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.

翻译：基于黎曼流形模型，我们研究了一种广泛应用的元监督学习算法——局部线性嵌入（LLE）在点云采样自紧致光滑带边界流形时的渐近行为。我们揭示了LLE在边界附近若干不同于基于扩散算法的特殊性质。特别地，我们证明了LLE逐点收敛于一个具有退化性的混合型微分算子，并计算了其收敛速率。文中讨论了算子双曲部分的影响，并提出了一种截断LLE算法，该算法是恢复狄利克雷拉普拉斯-贝尔特拉米算子的潜在途径。