信息几何与控制理论
Information Geometry and Control Theory

作者: 韩希武 , 孙华飞 :北京理工大学数学与统计学院,北京 ; 张真宁 :北京工业大学数理学院,北京 ;

关键词: 信息几何矩阵信息几何Fisher信息矩阵Lie群Information Geometry Matrix Information Geometry Fisher Information Matrix Lie Group

摘要:
本文首先简要介绍信息几何的基本内容,包含随机的情形和非随机的情形。通过引入Fisher信息矩阵、对偶联络的引入,来处理随机的统计流形;通过利用一般线性群的李子群以及子流形的理论,建立矩阵信息几何理论。然后介绍信息几何在控制理论中的应用,包含随机的情形和非随机的情形。

Abstract: In this paper, we first introduce briefly the fundamental contents of information geometry. By in-troducing the Fisher information matrix and the dual connections, we can deal with the statistical manifold. At the same time, by using the theory of Lie groups and submanifolds of the general linear group, the theory of matrix information geometry is constructed. Then we introduce the ap-plications of information geometry to control theory, including the random case and the non- random case.

文章引用: 韩希武 , 孙华飞 , 张真宁 (2016) 信息几何与控制理论。 动力系统与控制, 5, 135-142. doi: 10.12677/DSC.2016.54015

参考文献

[1] Amari, S. (1985) Differential-Geometrical Methods in Statistics. Springer Lectures Notes in Statistics, 28, 11-65.
http://dx.doi.org/10.1007/978-1-4612-5056-2_2

[2] Amari, S. and Nagaoka, H. (2007) Methods of Information Geometry. American Mathematical Soc, Vol. 191.

[3] Barbaresco, F. (2008) Innovative Tools for Radar Signal Processing Based on Cartan’s Geometry of SPD Matrices & Information Geometry. 2008 IEEE Radar Conference. IEEE.
http://dx.doi.org/10.1109/RADAR.2008.4720937

[4] Barbaresco, F. (2009) Interactions between Symmetric Cone and Infor-mation Geometries: Bruhat-Tits and Siegel Spaces Models for High Resolution Autoregressive Doppler Imagery, Emerging Trends in Visual Computing. Springer, Berlin, Heidelberg.

[5] Pennec, X., Fillard, P. and Ayache, N. (2006) A Riemannian Framework for Tensor Computing. International Journal of Computer Vision, 66, 41-66.
http://dx.doi.org/10.1007/s11263-005-3222-z

[6] Du, S., Zheng, N., Ying, S. and Liu, J. (2010) Affine Iterative Closest Point Algorithm for Point Set Registration. Pattern Recognition Letters, 31, 791-799.
http://dx.doi.org/10.1016/j.patrec.2010.01.020

[7] Du, S., Zheng, N., Xiong, L., Ying, S. and Xue, J. (2010) Scaling Iterative Closest Point ALGORITHM for Registration of m–D Point Sets. Journal of Visual Communication and Image Representation, 21, 442-452.
http://dx.doi.org/10.1016/j.jvcir.2010.02.005

[8] 黎湘, 程永强, 王宏强, 秦玉亮.雷达信号处理的信息几何[M]. 北京: 科学出版社, 2013.

[9] 黎湘, 程永强, 王宏强, 秦玉亮. 信息几何理论与应用研究进展[J]. 中国科学, 43(2013): 707-732.

[10] Peng, L., Sun, H., Sun, D. and Yi, J. (2011) The Geometric Structures and Instability of Entropic Dynamical Models. Advances in Mathematics, 227, 459-471.
http://dx.doi.org/10.1016/j.aim.2011.02.002

[11] Peng, L., Sun, H. and Xu, G. (2012) Information Geometric Characterization of the Complexity of Fractional Brownian Motions. Journal of Mathematical Physics, 53, 123305.
http://dx.doi.org/10.1063/1.4770047

[12] Li, C., Sun, H. and Zhang, S. (2013) Characterization of the Complexity of an ED Model via Information Geometry. The European Physical Journal Plus, 128, 1-6.
http://dx.doi.org/10.1140/epjp/i2013-13070-8

[13] Hall, B.C. (2015) Lie Groups, Lie Algebras, and Representations: An Ele-mentary Introduction (Vol. 222). Springer, New York.
http://dx.doi.org/10.1007/978-3-319-13467-3

[14] 孙华飞, 张真宁, 彭林玉, 段晓敏. 信息几何导引[M]. 北京: 科学出版社, 2016.

[15] Nielsen, F. and Bhatia, R. (2013) Matrix Information Geometry. Springer, New York.
http://dx.doi.org/10.1007/978-3-642-30232-9

[16] Moakher, M. (2005) A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices. SIAM Journal on Matrix Analysis and Applications, 26, 735-747.
http://dx.doi.org/10.1137/S0895479803436937

[17] Moakher, M. and Zéraï, M. (2011) The Riemannian Geometry of the Space of Positive-Definite Matrices and Its Application to the Regularization of Positive-Definite Matrix-Valued Data. Journal of Mathe-matical Imaging and Vision, 40, 171-187.
http://dx.doi.org/10.1007/s10851-010-0255-x

[18] Fiori, S. (2011) Solving Mini-mal-Distance Problems over the Manifold of Real-Symplectic Matrices. SIAM Journal on Matrix Analysis and Applications, 32, 938-968.
http://dx.doi.org/10.1137/100817115

[19] Fiori S. (2011) Extended Hamiltonian Learning on Riemannian Manifolds: Theoretical Aspects. IEEE Transactions on Neural Networks, 22, 687-700.
http://dx.doi.org/10.1109/TNN.2011.2109395

[20] Fiori S. (2012) Extended Hamiltonian Learning on Riemannian Manifolds: Numerical Aspects. IEEE Transactions on Neural Networks and Learning Systems, 23, 7-21.
http://dx.doi.org/10.1109/TNNLS.2011.2178561

[21] Dodson, C.T.J. and Wang, H. (2001) Iterative Approximation of Statistical Distributions and Relation to Information Geometry. Statistical Inference for Stochastic Processes, 4, 307-318.
http://dx.doi.org/10.1023/A:1012289028897

[22] Zhang, Z., Sun, H. and Zhong, F. (2009) Natural Gradient-Projection Algo-rithm for Distribution Control. Optimal Control Applications and Methods, 30, 495-504.
http://dx.doi.org/10.1002/oca.874

[23] Zhang Z., Sun, H. and Peng, L. (2013) Natural Gradient Algorithm for Stochastic Dis-tribution Systems with Output Feedback. Differential Geometry and Its Applications, 31, 682-690.
http://dx.doi.org/10.1016/j.difgeo.2013.07.004

[24] Zhang, Z., Sun, H., Peng, L. and Jiu, L. (2014) A Natural Gradient Algorithm for Stochastic Distribution Systems. Entropy, 16, 4338-4352.
http://dx.doi.org/10.3390/e16084338

[25] Zhong, F., Sun, H. and Zhang, Z. (2008) An Information Geometry Algorithm for Distribution Control. Bulletin of the Brazilian Mathematical Society, 39, 1-10.
http://dx.doi.org/10.1007/s00574-008-0068-3

[26] Duan, X., Sun, H. and Zhang, Z. (2014) A Natural Gradient Algorithm for the Solution of Lyapunov Equation Based on the Geodesic Distance. Journal of Computational Mathematics, 32, 93-106.
http://dx.doi.org/10.4208/jcm.1310-m4225

[27] Duan, X., Sun, H., Peng, L. and Zhao, X. (2013) A Natural Gradient Descent Algorithm for the Solution of Discrete Algebraic Lyapunov Equations Based on the Geodesic Distance. Applied Mathematics and Computation, 219, 9899- 9905.
http://dx.doi.org/10.1016/j.amc.2013.03.119

[28] Luo, Z. and Sun, H. (2014) Extended Hamil-tonian Algorithm for the Solution of Discrete Algebraic Lyapunov Equations. Applied Mathematics and Computation, 234, 245-252.
http://dx.doi.org/10.1016/j.amc.2014.02.037

[29] Luo, Z., Sun, H. and Duan, X. (2014) The Extended Hamiltonian Algorithm for the Solution of the Algebraic Riccati Equation. Journal of Applied Mathematics, 2014, Article ID: 693659.
http://dx.doi.org/10.1155/2014/693659

分享
Top