Intrinsic Grassmann Averages for Online Linear, Robust and Nonlinear Subspace Learning.

Area of Science:

• Machine Learning
• Computational Geometry
• Data Science

Background:

• Principal Component Analysis (PCA) and Kernel PCA (KPCA) are essential for dimensionality reduction.
• KPCA operates in infinite-dimensional spaces (reproducing Kernel Hilbert spaces - RKHS).
• Existing methods can be computationally intensive.

Purpose of the Study:

• To develop a geometric framework for dimensionality reduction.
• To compute principal linear subspaces in finite and infinite dimensions, including robust PCA.
• To create a faster and online version of KPCA.

Main Methods:

• Utilizing a geometric framework based on the Grassmann manifold.
• Computing the intrinsic average of subspaces spanned by observations.
• Developing an efficient algorithm for projection onto the average subspace.

Main Results:

• The geometric framework successfully computes principal components and extends to RKHS.
• A novel, efficient algorithm akin to KPCA is developed, offering substantial speed improvements.
• A new online version of KPCA is introduced.

Conclusions:

• The proposed geometric approach provides a faster and more efficient alternative to traditional KPCA.
• The developed algorithms demonstrate competitive performance on diverse datasets.
• This work offers significant advancements in dimensionality reduction techniques.