Seminar Details
2012-10-16 (14.00) : Averaging on Riemannian Manifolds: The Curses of Curvature and Topology
At Euler building
Organized by Mathematical Engineering
Speaker :
Bijan AFSARI (Center for Imaging Science, Johns Hopkins University)
Abstract :
A Riemannian manifold differs from a Euclidean space in its (non-trivial) curvature and topology. Current methods for defining notions of average for manifold-valued data extend
existing definitions from a Euclidean space to a Riemannian manifold. In this talk, we con-
sider some of these definitions and examine to what extent the curvature and topology of
the underlying manifold cause the corresponding averages to behave differently from their
Euclidean counterparts. We will start with very basic definitions and some historical re-
marks. The main theme of the talk is that the curses of curvature and topology, in one way
or another, haunt averaging on manifolds. In that regard, we consider both the inherent
properties (e.g., existence and uniqueness, noise sensitivity, continuity, etc) as well as algorithmic/computational properties associated with these averages. For example, we argue
that averaging on a Riemannian manifold can become increasingly difficult as the curvature
increases (in absolute value). As another example, we define a property called the averaging
property, which indicates how contaminating a set of data points with noise would change
their average; and we show that on a positively curved manifold, Riemannian averaging can
amplify the noise. We also study the adverse consequence of this fact on the smoothness
properties of recursive-iterative averages on such manifolds. Recursive-iterative averages
represent interesting examples of dynamical systems on manifolds and can be related to con-
sensus algorithms on manifolds. The global behavior of these systems is complicated since
both the curvature and topology of the manifold play roles. We study this global behavior
in the example of cyclic pursuit. Throughout the talk, we will consider numerous examples,
introduce the latest available results, and allude to open problems.
This talk is partly based on the speakers Ph.D. thesis under the supervision of P. S. Krishnaprasad and K. Grove at the University of Maryland. The rest of the talk is based on the joint work with R. Tron and R. Vidal at JHU. The presented research has been supported, in part, by grants from NSF, NIH, and ARL.