Principal Component Analysis in Space Forms

Area of Science:

• Data Science
• Differential Geometry
• Machine Learning

Background:

• Principal Component Analysis (PCA) is standard for Euclidean data.
• Hierarchical and cyclic data require non-Euclidean geometry.
• Dimensionality reduction on manifolds is challenging.

Purpose of the Study:

• Develop a novel PCA for non-Euclidean spaces (space forms).
• Introduce Space Form PCA (SFPCA) for manifold-valued data.
• Improve upon existing iterative dimensionality reduction methods.

Main Methods:

• Define PCA within constant curvature spaces (space forms).
• Utilize Riemannian affine subspaces for dimensionality reduction.
• Propose cost functions solvable via eigenequations for nested subspaces.

Main Results:

• SFPCA finds optimal low-dimensional affine subspaces.
• The method exhibits properties ensuring nested subspaces across dimensions.
• Evaluated on spherical and hyperbolic spaces with real and simulated data.

Conclusions:

• SFPCA outperforms existing methods in accuracy and convergence speed.
• Demonstrates superior performance in estimating true subspaces.
• Offers a theoretically sound and efficient alternative for manifold data analysis.