Comparing sets of data sets on the Grassmann and flag manifolds with applications to data analysis in high and low dimensions
Date
2020
Authors
Ma, Xiaofeng, author
Kirby, Michael, advisor
Peterson, Chris, advisor
Chong, Edwin, committee member
Scharf, Louis, committee member
Shonkwiler, Clayton, committee member
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Abstract
This dissertation develops numerical algorithms for comparing sets of data sets utilizing shape and orientation of data clouds. Two key components for "comparing" are the distance measure between data sets and correspondingly the geodesic path in between. Both components will play a core role which connects two parts of this dissertation, namely data analysis on the Grassmann manifold and flag manifold. For the first part, we build on the well known geometric framework for analyzing and optimizing over data on the Grassmann manifold. To be specific, we extend the classical self-organizing mappings to the Grassamann manifold to visualize sets of high dimensional data sets in 2D space. We also propose an optimization problem on the Grassmannian to recover missing data. In the second part, we extend the geometric framework to the flag manifold to encode the variability of nested subspaces. There we propose a numerical algorithm for computing a geodesic path and distance between nested subspaces. We also prove theorems to show how to reduce the dimension of the algorithm for practical computations. The approach is shown to have advantages for analyzing data when the number of data points is larger than the number of features.
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Subject
geodesic distance
Grassmann manifold
optimization
geometric data analysis
flag manifold
missing data