信息几何理论与应用研究进展

信息几何是在Riemann流形上采用现代微分几何方法来研究统计学问题的基础性、前沿性学科，被誉为是继Shannon开辟现代信息理论之后的又一新的理论变革，在信息科学与系统理论研究领域展现出了巨大的发展潜力．本文首先从参数化概率分布族的内蕴几何结构特征与信息的几何性质出发，精炼了信息几何的科学内涵，指出信息几何相比于经典统计学与信息论的理论优势与方法的革新．然后简要阐述了信息几何与微分几何的联系，综述了信息几何理论的发展历史与近20年来信息几何在神经网络、统计推断、通信编码、系统理论、物理学和医学成像等各领域应用的研究现状，归纳和总结了其中所体现的信息几何的基本原理和基本方法，并对信息几何的发展给予注记．特别地，对信息几何在信号处理领域中的应用成果进行了较全面的总结和概括，阐述了信息几何在信号检测、参数估计与滤波等方面的最新研究成果．最后，展望信息几何的发展前景，提出了信息几何在信号处理领域中的若干开放性问题．

Progress in theory and applications of information geometry

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 geometricM 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.