Applications of Differential Geometry to Signal Processing

Organizers: Jonathan H. Manton (The University of Melbourne) and Victor Barroso (Instituto Superior Tecnico)

In loose terms, differential geometry is the generalization of the machinery of differential calculus in Euclidean (flat) spaces to curved spaces. Curved spaces, or manifolds as they are formally known, are spaces which look locally like Euclidean space, but not globally. Simple examples of manifolds include the sphere and the torus (doughnut).

Analogous to linear algebra providing a systematic approach to linear signal processing problems, differential geometry provides a systematic approach to an important class of non-linear signal processing problems. Subspace tracking is but one example of a signal processing problem naturally handled by differential geometry, in this case because the collection of all subspaces of a certain dimension is a Grassmann manifold.

The talks will demonstrate by example how differential geometry can be applied to a broad range of signal processing problems.

Overview lecture:
  Author: Jonathan H. Manton (The University of Melbourne)
  Title: On the Role of Differential Geometry in Signal Processing

Regular lectures:

• Title: Quantization on the Grassmann manifold: Applications to precoded MIMO wireless systems
  Authors: Bishwarup Mondal, University of Texas, Austin; Robert Heath, University of Texas, Austin; Hanlen Leif, National ICT Australia
• Title: Applications of Planar Shape Analysis to Image-Based Inferences
  Authors: Anuj Srivastava, Florida State University; Shantanu Joshi, Florida State University; David Kaziska, Florida State University; David Wilson, University of Florida
• Title: Information Geometry of Turbo and LDPC Codes
  Author: Shiro Ikeda, Institute of Statistical Mathematics
• Title: Intrinsic Variance Lower Bound (IVLB): An Extension of the Cramer-Rao Bound to Riemannian Manifolds
  Authors: Joao Xavier, Instituto Superior Tecnico - ISR; Victor Barroso, Instituto Superior Tecnico - ISR
• Title: Generalizations of the Rayleigh Quotient Iteration for the Iterative Refinement of the Eigenvectors of Real Symmetric Matrices
  Authors: Maziar Nikpour, Catholic University of Louvain; Knut Hueper, National ICT Australia; Jonathan Manton, The University of Melbourne