Grassmann, Flag, and Schubert varieties in applications

Abstract

This dissertation develops mathematical tools for signal processing and pattern recognition tasks where data with the same identity is assumed to vary linearly. We build on the growing canon of techniques for analyzing and optimizing over data on Grassmann manifolds. Specifically we expand on a recently developed method referred to as the flag mean that finds an average representation for a collection data that consists of linear subspaces of possibly different dimensions. When prior knowledge exists about relationships between these data, we show that a point analogous to the flag mean can be found as an element of a Schubert variety to incorporates this theoretical information. This domain restriction relates closely to a recent result regarding point-to-set functions. This restricted average along with a property of the flag mean that prioritizes weak but common information, leads to practical applications of the flag mean such as chemical plume detection in long-wave infrared hyperspectral videos, and a modification of the well-known diffusion map for adaptively visualizing data relationships in 2-dimensions.