Local covariance structure under the manifold setup has been widely applied
in the machine-learning community. Based on the established theoretical
results, we provide an extensive study of two relevant manifold learning
algorithms, empirical intrinsic geometry (EIG) and 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.
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