Classification on the Grassmannians: theory and applications
Date
2008
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Abstract
This dissertation consists of four parts. It introduces a novel geometric framework for the general classification problem and presents empirical results obtained from applying the proposed method on some popular classification problems. An analysis of the robustness of the method is provided using matrix perturbation theory, which in turn motivates an optimization problem to improve the robustness of the classifier. Lastly, we illustrate the use of compressed data representations based on Karcher mean.
The success of this geometric framework builds upon the facts that the geometry and statistics of the Grassmannians are well-understood and families of patterns with a common characterization possesses discriminatory variations that are useful for classification. Under the right conditions, these families of patterns can be viewed as points on the Grassmannian where distances are available for classification. In this dissertation, we will make precise this connection, review various ways these metrics arise, and how to efficiently compute distances between points on this manifold.
Under this framework, we achieve excellent classification results for a variety of applications in face recognition and offer new insights to the problem in general. Attempting to break the method, we consider nonlinear data sets and images of extremely low resolutions. We are pleased to learn that the Grassmann method is robust against resolution reductions.
In order to understand how robust the Grassmann method is against perturbation, we draw support from matrix perturbation theory where we exploit the natural correspondence between linear subspaces and points on the Grassmannians. We are then led to formulate an optimization problem using these characteristics as an objective function and further connect this optimization criterion to the idea of Fisher Linear Discriminant on general image sets. Numerical solutions obtained show promising improvements on the separability criterion.
The thesis is concluded by providing a novel algorithm that computes subject prototypical points using the Karcher mean on the Grassmannian. A lot of new ideas for geometric data analysis are generated through studies of old ideas. We hope that the suite of these frameworks and algorithms can collectively provide useful insights in studying geometric aspects of large data sets.
The success of this geometric framework builds upon the facts that the geometry and statistics of the Grassmannians are well-understood and families of patterns with a common characterization possesses discriminatory variations that are useful for classification. Under the right conditions, these families of patterns can be viewed as points on the Grassmannian where distances are available for classification. In this dissertation, we will make precise this connection, review various ways these metrics arise, and how to efficiently compute distances between points on this manifold.
Under this framework, we achieve excellent classification results for a variety of applications in face recognition and offer new insights to the problem in general. Attempting to break the method, we consider nonlinear data sets and images of extremely low resolutions. We are pleased to learn that the Grassmann method is robust against resolution reductions.
In order to understand how robust the Grassmann method is against perturbation, we draw support from matrix perturbation theory where we exploit the natural correspondence between linear subspaces and points on the Grassmannians. We are then led to formulate an optimization problem using these characteristics as an objective function and further connect this optimization criterion to the idea of Fisher Linear Discriminant on general image sets. Numerical solutions obtained show promising improvements on the separability criterion.
The thesis is concluded by providing a novel algorithm that computes subject prototypical points using the Karcher mean on the Grassmannian. A lot of new ideas for geometric data analysis are generated through studies of old ideas. We hope that the suite of these frameworks and algorithms can collectively provide useful insights in studying geometric aspects of large data sets.
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Subject
classification
face recognition
geometric data analysis
Grassmannians
manifolds
set-to-set
mathematics
computer science