Boundary Detection Algorithm Inspired by Locally Linear Embedding

TL;DR

This paper introduces a boundary detection algorithm for high-dimensional data based on locally linear embedding, utilizing local covariance analysis and noise robustness, with demonstrated effectiveness on simulated data.

Contribution

It presents a novel boundary detection method inspired by locally linear embedding, incorporating spectral analysis and noise handling, advancing boundary identification in high-dimensional data.

Findings

Effective boundary detection in simulated high-dimensional data

Spectral analysis guides parameter selection

Enhanced robustness to high-dimensional noise

Abstract

In the study of high-dimensional data, it is often assumed that the data set possesses an underlying lower-dimensional structure. A practical model for this structure is an embedded compact manifold with boundary. Since the underlying manifold structure is typically unknown, identifying boundary points from the data distributed on the manifold is crucial for various applications. In this work, we propose a method for detecting boundary points inspired by the widely used locally linear embedding algorithm. We implement this method using two nearest neighborhood search schemes: the epsilon-radius ball scheme and the K-nearest neighbor scheme. This algorithm incorporates the geometric information of the data structure, particularly through its close relation with the local covariance matrix. We analyze the algorithm by exploring the spectral properties of the local covariance matrix, with the findings guiding the selection of key parameters. In the presence of high-dimensional noise, we propose a framework aimed at enhancing boundary detection in noisy data. Furthermore, we demonstrate the algorithm's performance with simulated examples.