信息几何研究进展  

The Research Progress of Information Geometry

作　　者：孙华飞 曾澍楠 SUN Hua-fei;ZENG Shu-nan(School of Mathematics and Statistics,Beijing Institute of Technology,Beijing 100081,China)

机构地区：[1]北京理工大学数学与统计学院,北京100081

出　　处：《科学技术与工程》2020年第30期12247-12254,共8页Science Technology and Engineering

基　　金：北京市科技计划(Z161100005016043)。

摘　　要：随着人工智能的不断深入,基于欧氏框架的数学理论无法有效解决信息领域中的一些非线性和随机性问题,而信息几何是解决非线性和随机性问题的有效工具。基于黎曼几何的信息几何由于其在统计推断、信号处理、图像处理、神经网络、机器学习等领域的广泛应用,受到了人们的关注,成为热门的研究领域。经过几十年的发展,信息几何已经从最初鲜为人知的领域发展成为研究非线性、随机性复杂信息的重要工具。对信息几何研究进展进行综述,首先介绍信息几何的理论框架,包括对偶联络、流形上的测地距离及黎曼梯度等,然后简要介绍信息几何在统计推断、神经网络、控制系统领域、信号处理和机器学习等领域的应用,最后展望信息几何的研究方向。With the deepening of artificial intelligence,the mathematical theory based on Euclidean framework is unable no solve effectively some nonlinear and stochastic problems in the information field,while information geometry is an effective tool to solve those problems.Information geometry is based on Riemannian geometry and widely used in statistical inference,signal processing,image processing,neural network,machine learning,and other fields,which has attracted people's attention and becomes a hot research field.After decades of development,information geometry has become an important tool for the study of nonlinear,stochastic,and complex information.The research progress of information geometry is reviewed.First,the theoretical framework of information geometry is introduced,including dual connection,geodesic distance on manifold,and Riemann gradient,etc.,then the application of information geometry in the fields of statistical inference,neural network,control system,signal processing,machine learning,etc.,is briefly presented,and finally the future of information geometry is prospected.This review shall help readers interested in information geometry in understanding the basic theoretical framework and the important application scenarios of information geometry,and provide some inspiration for solving the bottleneck problems in the information field.

关 键 词：信息几何 黎曼梯度 黎曼度量 测地距离 李群与李代数 

分 类 号：O186[理学—数学]