NEWS in the Department of Mathematics
Anthony (Drew) Schwickerath - PhD defense
Date: Thursday, October 23, 2014
Time: 10:00 a.m.
Place: Weber 015
TITLE: Linear Models, Signal Detection, and the Grassmann Manifold
Dr. Michael Kirby
Dr. Chris Peterson
Dr. Louis Scharf
Dr. Richard Eykholt
Traditionally, signal processing has focused on data living in a vector space, typically real or complex space. Commonly used methods such as matched subspace detectors assume a standard linear model and are built upon a sound statistical basis. The associated algorithms are well-behaved, simple to implement, and, as a result, widely used.
Rather than using raw data, other methods aggregate data as a point on a Grassmann manifold. The geometric intuition is appealling; however, examples we have found in the literature which utilize the Grassmann manifold lack the principled rigor and general applicability of their vector space counterparts. We believe that the lack of an appropriate theoretical framework hinders the maturation of signal processing on Grassmann manifolds as a field.
To rectify this, we propose a geometric framework on the Grassmann manifold. In our framework, a generalization of traditional linear subspace models produces a family of Schubert varieties on the same Grassmann manifold as our aggregate data resides. For a broad class of orthogonally invariant functions on pairs of points, we develop analogues of the Euclidean point-to-plane distance and projection of a point to a plane. Simple algorithms are then presented which apply these results to the problems of signal detection and recovery.