| Abstract: The de Bruijn’s identity is a classical result in Information. Theory relating two well-known measures of information in distributions: Shannon Entropy and Fisher Information. It measures, in terms of the Fisher Information, how the entropy of a distribution increases as it evolves under Brownian noise. The original identity was proved for distributions on the real line, but is not hard to extend to n dimensions.
Our work has attempted to find analogues in the context of Riemannian manifolds. There, it is not even clear, ab initio, what the statement of the theorem should be. Our approach requires extension of the notion of Intrinsic Fisher Information to Riemannian manifolds. From this we are able to show that a probabilistic solution to the heat equation results in a heat equation for the entropy density of this solution, but with a source term that arises from the Fisher Information density. Combined with the Li-Yau inequality this lead to bounds on the rate of increase of entropy on a manifold. The seminar will discuss ideas around Fisher Information, forms on manifolds, and the heat equation, as well as a little history of the identity. This is joint work with Stephen Howard (DST Edinburgh), Doug Cochran (Arizona State U).
Bio: Professor Bill Moran currently is in the Department of Electrical and Electronic Engineering at The University of Melbourne. Previously he served as the Director of Signal Processing and Sensor Control Group in the School of Engineering, at RMIT and before that Director of the Defence Science Institute (2011–2014) in the University of Melbourne, Professor of Mathematics (’76–’91), Head of the Department of Pure Mathematics (’77–’79, ’84–’86), Dean of Mathematical and Computer Sciences (’81, ’82, ’89) at the University of Adelaide, and Head of the Mathematics Discipline at the Flinders University of South Australia (’91–’95), during which time he worked in various roles in CSSIP. He was a member of the Australian Research Council College of Experts (2007-09). He was elected to the Fellowship of the Australian Academy of Science in 1984. He holds a Ph.D. in Pure Mathematics from the University of Sheffield, UK (’68), and a First Class Honours B.Sc. in Mathematics from the University of Birmingham (’65). He has been a Principal Investigator on numerous research grants and contracts, in areas spanning pure mathematics to radar development, from both Australian and US Research Funding Agencies, including DARPA, AFOSR, AFRL, Australian Research Council (ARC), and DSTO. His main areas of research interest are in signal processing both theoretically and in applications to radar, waveform design and radar theory, sensor networks, and sensor management. He also works in various areas of mathematics including harmonic analysis, representation theory, and number theory. |
