  A novel method, named Curvature-Augmented Manifold Embedding and Learning
(CAMEL), is proposed for high dimensional data classification, dimension
reduction, and visualization. CAMEL utilizes a topology metric defined on the
Riemannian manifold, and a unique Riemannian metric for both distance and
curvature to enhance its expressibility. The method also employs a smooth
partition of unity operator on the Riemannian manifold to convert localized
orthogonal projection to global embedding, which captures both the overall
topological structure and local similarity simultaneously. The local orthogonal
vectors provide a physical interpretation of the significant characteristics of
clusters. Therefore, CAMEL not only provides a low-dimensional embedding but
also interprets the physics behind this embedding. CAMEL has been evaluated on
various benchmark datasets and has shown to outperform state-of-the-art
methods, especially for high-dimensional datasets. The method's distinct
benefits are its high expressibility, interpretability, and scalability. The
paper provides a detailed discussion on Riemannian distance and curvature
metrics, physical interpretability, hyperparameter effect, manifold stability,
and computational efficiency for a holistic understanding of CAMEL. Finally,
the paper presents the limitations and future work of CAMEL along with key
conclusions.
