摘要
信息几何是近年来快速发展起来的前沿性基础学科,是在Riemann流形上利用现代微分几何方法研究统计学领域问题而提出的一套新的理论体系.同时,信息几何作为一个具有深刻理论基础的跨学科研究课题,不仅为解决统计学问题引入新的观念,也为系统理论、信息科学、神经网络和统计推断等领域的研究提供新的方法.正是由于信息几何在解决相关领域科学问题时所拥有的强大的理论和技术优势,吸引着众多学者不断探索信息几何在自然科学各个领域的潜在应用.文章首先简要介绍了信息几何理论的一些基本概念,包括统计流形及其上的黎曼度规、仿射联络以及散度的概念,并介绍了对偶平坦流形上两个重要定理;然后,综述了国内外关于信息几何理论在神经网络、统计推断、系统控制理论、物理学、通信编码和医学成像等领域的最新应用进展.最后对信息几何的研究前景进行了展望并总结全文.
Information geometry is the fundamental and cutting-edge discipline rapidly developed in recent years,which is a new set of theoretical system that explores statistical problems on Riemannian manifolds of probability distributions by using the methods of modern differential geometry.Meanwhile,as an interdisciplinary research subject with profound theoretical foundation,information geometry theory provides a new concept for solving statistical problems and a new approach for studying systems theory,information science,neural networks,statistical inference and other research fields.Just because of powerful theory and technology advantages in solving scientific problems with information geometry method,which attracts a lot of researchers and results in a great many of studies in various fields of natural sciences.Firstly,this article introduces some basic concepts of information geometry,including statistical manifold,Riemannian metric,affine connection and Kullback-Leibler divergence.The two important theorems based on dual flat manifold are also introduced.Then,the new application of information geometry to various areas such as neural networks,statistical inference,systems theory,communications and coding,physics and medical imaging are reviewed.And the research prospect of information geometry is also discussed in the last.
出处
《渤海大学学报（自然科学版）》
CAS
2017年第3期199-205,共7页
Journal of Bohai University:Natural Science Edition
基金
宝鸡文理学院重点资助项目(No:ZK15078)
