In this paper, we propose a fast and effective neural network algorithm to perform singular value decomposition (SVD) of a cross-covariance matrix between two high-dimensional data streams.Firstly, we derive a dynamical system from a newly proposed information criterion.The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms.It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.
In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms.It is shown how an O(n 2 ) SVD updating algorithm can restore an acceptable approximation at every stage, with a fairly small tracking error of approximately the time variation in O(n) time steps.Finally, an error analysis is performed, proving that the algorithm is stable, when supplemented with a Jacobi-type re-orthogonalization procedure, which can easily be incorporated into the updating scheme. His current research interests include Radar Signal Processing, Blind Source Separation and Broadband Wireless Communication Technology.Jianqiang Qin received the Bachelor’s and Master’s degree from Xi’an Research Institute of High Technology, Xi’an, Shaanxi, China, in 20, respectively, both in electrical engineering. In this paper, we extend the well known QR-updating scheme to a similar but more versatile and generally applicable scheme for updating the singular value decomposition (SVD).This is done by supplementing the QR-updating with a Jacobi-type SVD procedure, where apparently only a few SVD steps after each QR-update suffice in order to restore an acceptable approximation for the SVD.Traditionally, the singular value decomposition (SVD) has been used in rank and subspace tracking methods.However, the SVD is computationally costly, especially when the problem is recursive in nature and the size of the matrix is large.The truncated ULV decomposition (TULV) is an alternative to the SVD. The algorithm is most efficient when the matrix is of low rank.It provides a good approximation to subspaces for the data matrix and can be modified quickly to reflect changes in the data. Numerical results are presented that illustrate the accuracy of the algorithm.