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. Particularly, LLE converges to a mixed-type differential operator with degeneracy. This study leads to an alternative boundary detection algorithm and two potential approaches to recover the Dirichlet Laplace-Beltrami operator.

On the spectral property of kernel-based sensor fusion algorithms of high dimensional data

“…This viewpoint leads to a natural question -if the kernel is not symmetric, how much can we say about the spectrum of the kernel-based affinity matrix? This problem is not unique to the sensor fusion problem, and appears more naturally in other algorithms, like the locally linear embedding (LLE) [62]. In the LLE, the established "affinity matrix" is in general not symmetric due to the asymmetric data geometric structure.…”

“…This naturally links the LLE to an asymmetric kernel. See Figure 2.1 in [62] for an example of the spectrum of the LLE under the null case. Fourth, the whole argument in the current paper can be carried over to the subgaussian, or more general setup.…”

Convergence analysis of Adaptive Locally Iterative Filtering and SIFT method

“…Specifically, the kernel associated with LLE is determined by the data, and its bandwidth depends on the point. Moreover, the kernel morphology near the boundary is even different from that away from the boundary [51]. We refer the interested readers to [50,51] for details.…”

“…Moreover, the kernel morphology near the boundary is even different from that away from the boundary [51]. We refer the interested readers to [50,51] for details.…”