  Local covariance structure under the manifold setup has been widely applied
in the machine learning society. Based on the established theoretical results,
we provide an extensive study of two relevant manifold learning algorithms,
empirical intrinsic geometry (EIG) and the locally linear embedding (LLE) under
the manifold setup. Particularly, we show that without an accurate dimension
estimation, the geodesic distance estimation by EIG might be corrupted.
Furthermore, we show that by taking the local covariance matrix into account,
we can more accurately estimate the local geodesic distance. When understanding
LLE based on the local covariance structure, its intimate relationship with the
curvature suggests a variation of LLE depending on the "truncation scheme". We
provide a theoretical analysis of the variation.
