Nested Grassmannians for Dimensionality Reduction with Applications.

Area of Science:

• Differential Geometry
• Manifold Learning
• Data Science

Background:

• Dimensionality reduction techniques are crucial for analyzing complex data.
• Principal Geodesic Analysis (PGA) is a popular method, but alternatives are being explored.
• Nested structures on Riemannian manifolds offer a new approach to dimensionality reduction.

Purpose of the Study:

• To propose a novel framework for constructing nested sequences of homogeneous Riemannian manifolds.
• To apply this framework to the Grassmann manifold, creating Nested Grassmannians (NG).
• To develop supervised and unsupervised dimensionality reduction algorithms using the NG structure for planar shape analysis.

Main Methods:

• Developing a framework that exploits the geometry of homogeneous Riemannian manifolds.
• Constructing nested lower-dimensional submanifolds that are not necessarily geodesic.
• Applying the framework to Grassmann manifolds for planar shape analysis.

Main Results:

• The proposed framework generates Nested Grassmannians (NG).
• Algorithms for supervised and unsupervised dimensionality reduction were developed based on NG.
• Simulations and real data experiments showed NG algorithms achieve a higher expressed variance ratio than PGA.

Conclusions:

• The proposed framework provides an effective method for constructing nested homogeneous Riemannian manifolds.
• Nested Grassmannians offer a valuable tool for dimensionality reduction in applications like planar shape analysis.
• The developed algorithms demonstrate superior performance compared to existing methods like PGA.