Advisor: Dr. Michael Kirby

Committee:  Dr. Emily King,Dr. Chris Peterson,Dr. Charles Anderson

Title: Subspace and Network Averaging for Computer Vision and Bioinformatics

Abstract: Finding a central prototype (a.k.a. average) from cluster of points may seem like a problem from elementary school statistics (e.g.

the mean), but has broad application to complex problems like action clustering in computer vision or gene co-expression module representation in bioinformatics. A central prototype of a set of points is the solution to an optimization problem that either minimizes distance or maximizes similarity between the prototype and each point in the cluster. In this paper we offer four novel prototypes for a cluster of points: the flag median, maximally correlated flag, cluster expression vector and eigengene subspace. We will formalize the flag median, and the maximally correlated flag using subspace representations for data, specifically the Grassmann and Flag manifolds. In addition to introducing these prototypes, we will derive a novel algorithm which can be used to calculate subspace prototypes- FlagIRLS. The third and fourth prototypes, the cluster expression vector and eigengene subspace, are inspired by problems involving gene cluster (e.g. pathway or module) representations. The cluster expression vector leverages connections within networks of genes whereas the eigengene subspace is computed using PCA (Principal Component Analysis). In this work we will explore the theoretical underpinnings of these prototypes, find algorithms to compute them, and apply them to computer vision and biological data sets.