  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 discuss the selection the key parameter and
analyze the algorithm through our exploration of the spectral properties of the
local covariance matrix in both neighborhood search schemes. Furthermore, we
demonstrate the algorithm's performance with simulated examples.
