信息几何理论与应用研究进展
摘要
Information geometry is the fundamental and cutting-edge discipline that explores statistical problems on Riemannian manifolds of probability distributions using the methods of differential geometry. It has been identified as the second generation of modern information theory pioneered by Shannon, and exhibits great potential in developing the field of information science and systems theory. This paper begins defining the scientific content of information geometry from the intrinsic geometrical structures of parameterized families of probability distributions as well as geometric properties of information; it points out theoretical advantages and methodological innovations of information geometry compared with classical statistics and information theory. Next, the paper briefly introduces connections between information geometry and differential geometry, and elaborates the history of information geometry as well as its applications to various areas such as neural networks, statistical inference, communications and coding, systems and control theory, physics and medical imaging, among others. In particular, the applications of information geometry to signal processing, including the latest results of geometric methods of signal detection, nonlinear parameter estimation, and filtering, are comprehensively introduced. The basic ideas and methods of information geometry are also summarized. Finally, by viewing prospects for the development of information geometry, several open problems of information geometry in applications to signal processing are proposed.