Curvature Augmented Manifold Embedding and Learning
Yongming Liu
A new dimensional reduction (DR) and data visualization method,
Curvature-Augmented Manifold Embedding and Learning (CAMEL), is proposed. The
key novel contribution is to formulate the DR problem as a mechanistic/physics
model, where the force field among nodes (data points) is used to find an
n-dimensional manifold representation of the data sets. Compared with many
existing attractive-repulsive force-based methods, one unique contribution of
the proposed method is to include a non-pairwise force. A new force field model
is introduced and discussed, inspired by the multi-body potential in
lattice-particle physics and Riemann curvature in topology. A
curvature-augmented force is included in CAMEL. Following this, CAMEL
formulation for unsupervised learning, supervised learning, semi-supervised
learning/metric learning, and inverse learning are provided. Next, CAMEL is
applied to many benchmark datasets by comparing existing models, such as tSNE,
UMAP, TRIMAP, and PacMap. Both visual comparison and metrics-based evaluation
are performed. 14 open literature and self-proposed metrics are employed for a
comprehensive comparison. Conclusions and future work are suggested based on
the current investigation. Related code and demonstration are available on
https://github.com/ymlasu/CAMEL for interested readers to reproduce the results
and other applications.