Abstract :

In recent years, techniques involving differential geometry have been gaining popularity in the signal processing community. One such area is optimisation on manifolds; a number of important signal processing problems can be reformulated as an optimisation problem on a smooth curved space, or more generally, on a manifold.

The traditional approach to developing a numerical algorithm for minimising a cost function on a manifold is to endow the manifold with a metric structure and then use judiciously parallel transport and the Riemannian exponential map to generalise Euclidean algorithms (such as the conjugate gradient or the Newton method) to the manifold setting. This talk will present a more general framework and explain the theoretical and practical advantages of working in this greater generality. Specifically, this framework enables an arbitrary algorithm in Euclidean space to be transported to a manifold, and is sufficiently flexible to allow the domains of attraction, the computational complexity and the convergence rate to be altered significantly, while at the same time guaranteeing the local convergence of the transported algorithm to be the same or faster than the original algorithm. The challenge of extending this framework to stochastic filtering on manifolds will be touched on informally.

Biography :

Professor Manton received his Bachelor of Science (mathematics) and Bachelor of Engineering (electrical) degrees in 1995 and his Ph.D. degree in 1998, all from the University of Melbourne, Australia. From 1998 to 2004, he was with the Department of Electrical and Electronic Engineering at the University of Melbourne. During that time, he held a Postdoctoral Research Fellowship then subsequently a Queen Elizabeth II Fellowship, both from the Australian Research Council. In 2005 he became a full Professor in the Department of Information Engineering, Research School of Information Sciences and Engineering (RSISE) at the Australian National University. From July 2006 till May 2008, he was on secondment to the Australian Research Council as Executive Director, Mathematics, Information and Communication Sciences. Currently, he holds a distinguished Chair at the University of Melbourne with the title Future Generation Professor. He is also an Adjunct Professor in the Mathematical Sciences Institute at the Australian National University. Professor Manton's traditional research interests range from pure mathematics (e.g. commutative algebra, algebraic geometry, differential geometry) to engineering (e.g. signal processing, wireless communications). Recently though, led by a desire to participate in the convergence of the life sciences and the mathematical sciences, he has commenced learning neuroscience.

Professor Manton has served recently as an Associate Editor for IEEE Transactions on Signal Processing, a Committee Member for IEEE Signal Processing for Communications (SPCOM) Technical Committee, and a Committee Member on the Mathematics Panel for the ACT Board of Senior Secondary Studies in Australia.