Subspace and network averaging for computer vision and bioinformatics

Abstract

Finding a central prototype (a.k.a. average) from a cluster of points in high dimensional space has broad applications 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 may be cast as the solution to an optimization problem that either minimizes distance or maximizes similarity between the prototype and each point in the cluster. In this dissertation 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 Grasmann 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 Principal Component Analysis (PCA). In this work we will explore the theoretical under-pinnings of these prototypes, find algorithms to compute and them to computer vision and biological data sets.