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### 互相关Hebbian算法的离散时间动力学特性分析

• 收稿日期:2020-05-21 修回日期:2021-07-20 接受日期:2021-12-29 出版日期:2022-01-28 发布日期:2022-01-28

### On the Discrete-Time Dynamics of Cross-Coupled Hebbian Algorithm

Xiao-Wei Feng (冯晓伟), Xiang-Yu Kong* (孔祥玉), Chuan He (何川), and Dong-Hui Xu (徐东辉)

1. Xi'an Research Institute of High Technology, Xi'an 710025, China
• Received:2020-05-21 Revised:2021-07-20 Accepted:2021-12-29 Online:2022-01-28 Published:2022-01-28
• Contact: Xiang-Yu Kong E-mail:xiangyukong01@163.com
• About author:Xiang-Yu Kong received his Bachelor's degree in control science and engineering from Beijing Institute of Technology, Beijing, in 1990, his Master's degree in control science and engineering from Xi'an Research Institute of High Technology, Xi'an, in 2000, and his Ph.D. degree in control science and engineering from the Xi'an Jiaotong University, Xi'an, in 2005. He is now a professor at Xi'an Research Institute of High Technology, Xi'an. He has authored or coauthored more than 100 refereed articles on IEEE Transactions on Signal Processing, IEEE Transactions on Neural Networks and Learning Systems and others journals, and has published five monographs, including one entitled Principal Component Analysis Networks and Algorithms (Springer, 2017). His research interests cover stochastic system analysis, nonlinear system modeling and its application and complex system fault diagnosis.
• Supported by:
The work was supported by the National Natural Science Foundation of China under Grant Nos.61903375, 61673387 and 61773389, the Natural Science Foundation of Shaanxi Province of China under Grant Nos.2020JM-356 and 2020JQ-298, and the Postdoctoral Science Foundation of China under Grant No.2019M663635.

Abstract: Principal/minor component analysis (PCA/MCA), generalized principal/minor component analysis (GPCA/GMCA), and singular value decomposition (SVD) algorithms are important techniques for feature extraction. In the convergence analysis of these algorithms, the deterministic discrete-time (DDT) method can reveal the dynamic behavior of PCA/MCA and GPCA/GMCA algorithms effectively. However, the dynamic behavior of SVD algorithms has not been studied quantitatively because of their special structure. In this paper, for the first time, we utilize the advantages of the DDT method in PCA algorithms analysis to study the dynamics of SVD algorithms. First, taking the cross-coupled Hebbian algorithm as an example, by concatenating the two cross-coupled variables into a single vector, we successfully get a PCA-like DDT system. Second, we analyze the discrete-time dynamic behavior and stability of the PCA-like DDT system in detail based on the DDT method, and obtain the boundedness of the weight vectors and learning rate. Moreover, further discussion shows the universality of the proposed method for analyzing other SVD algorithms. As a result, the proposed method provides a new way to study the dynamical convergence properties of SVD algorithms.

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