Low-Rank Riemannian Optimization Approach to the Role Extraction Problem

This dissertation uses Riemannian optimization theory to increase our understanding of the role extraction problem and algorithms. Recent ideas of using the low-rank projection of the neighborhood pattern similarity measure and our theoretical analysis of the relationship between the rank of the similarity measure and the number of roles in the graph motivates our proposal to use Riemannian optimization to compute a low-rank approximation of the similarity measure. We propose two indirect approaches to use to solve the role extraction problem. The first uses the standard two-phase process. For the first phase, we propose using Riemannian optimization to compute a low-rank approximation of the similarity of the graph, and for the second phase using k-means clustering on the low-rank factor of the similarity matrix to extract the role partition of the graph. This approach is designed to be efficient in time and space complexity while still being able to extract good quality role partitions. We use basic experiments and applications to illustrate the time, robustness, and quality of our two-phase indirect role extraction approach. The second indirect approach we propose combines the two phases of our first approach into a one-phase approach that iteratively approximates the low-rank similarity matrix, extracts the role partition of the graph, and updates the rank of the similarity matrix. We show that the use of Riemannian rank-adaptive techniques when computing the low-rank similarity matrix improves robustness of the clustering algorithm.