信息几何研究进展

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地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。将对信息几何研究进展做一个综述。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、以及黎曼梯度等，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、机器学习等领域的应用，最后介绍信息几何的展望，期望对信息几何感兴趣的学者有所帮助。通过该综述，读者可以了解到信息几何的基本理论框架，了解到信息几何的重要应用场景，为解决信息领域中的瓶颈问题提供一定的启发。

Research Progress of Information Geometry

With the deepening of artificial intelligence, the mathematical theory based on Euclidean framework can not effectively solve some nonlinear and stochastic problems in the information field, while information geometry is an effective tool to solve the nonlinear and stochastic problems. Information geometry based on Riemannian geometry is 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. In this paper, the research progress of information geometry is reviewed. This paper first introduces the theoretical framework of information geometry, including dual connection, geodesic distance on manifold, and Riemann gradient, etc., then briefly introduces the application of information geometry in the fields of statistical inference, neural network, control system, signal processing, machine learning, etc., and finally introduces the prospect of information geometry, which is expected to be helpful to the scholars interested in information geometry. Through this review, readers can understand the basic theoretical framework of information geometry and the important application scenarios of information geometry, which provides some inspiration for solving the bottleneck problems in the information field.