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Volume 11, Issue 4
Best Rank-One Approximation of Fourth-Order Partially Symmetric Tensors by Neural Network

Xuezhong Wang, Maolin Che & Yimin Wei

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 673-700.

Published online: 2018-06

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  • Abstract

Our purpose is to compute the multi-partially symmetric rank-one approximations of higher-order multi-partially symmetric tensors. A special case is the partially symmetric rank-one approximation for the fourth-order partially symmetric tensors, which is related to the biquadratic optimization problem. For the special case, we implement the neural network model by the ordinary differential equations (ODEs), which is a class of continuous-time recurrent neural network. Several properties of states for the network are established. We prove that the solution of the ODE is locally asymptotically stable by establishing an appropriate Lyapunov function under mild conditions. Similarly, we consider how to compute the multi-partially symmetric rank-one approximations of multi-partially symmetric tensors via neural networks. Finally, we define the restricted $M$-singular values and the corresponding restricted $M$-singular vectors of higher-order multi-partially symmetric tensors and design to compute them. Numerical results show that the neural network models are efficient.

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@Article{NMTMA-11-673, author = {}, title = {Best Rank-One Approximation of Fourth-Order Partially Symmetric Tensors by Neural Network}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {11}, number = {4}, pages = {673--700}, abstract = {

Our purpose is to compute the multi-partially symmetric rank-one approximations of higher-order multi-partially symmetric tensors. A special case is the partially symmetric rank-one approximation for the fourth-order partially symmetric tensors, which is related to the biquadratic optimization problem. For the special case, we implement the neural network model by the ordinary differential equations (ODEs), which is a class of continuous-time recurrent neural network. Several properties of states for the network are established. We prove that the solution of the ODE is locally asymptotically stable by establishing an appropriate Lyapunov function under mild conditions. Similarly, we consider how to compute the multi-partially symmetric rank-one approximations of multi-partially symmetric tensors via neural networks. Finally, we define the restricted $M$-singular values and the corresponding restricted $M$-singular vectors of higher-order multi-partially symmetric tensors and design to compute them. Numerical results show that the neural network models are efficient.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2018.s01}, url = {http://global-sci.org/intro/article_detail/nmtma/12467.html} }
TY - JOUR T1 - Best Rank-One Approximation of Fourth-Order Partially Symmetric Tensors by Neural Network JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 673 EP - 700 PY - 2018 DA - 2018/06 SN - 11 DO - http://doi.org/10.4208/nmtma.2018.s01 UR - https://global-sci.org/intro/article_detail/nmtma/12467.html KW - AB -

Our purpose is to compute the multi-partially symmetric rank-one approximations of higher-order multi-partially symmetric tensors. A special case is the partially symmetric rank-one approximation for the fourth-order partially symmetric tensors, which is related to the biquadratic optimization problem. For the special case, we implement the neural network model by the ordinary differential equations (ODEs), which is a class of continuous-time recurrent neural network. Several properties of states for the network are established. We prove that the solution of the ODE is locally asymptotically stable by establishing an appropriate Lyapunov function under mild conditions. Similarly, we consider how to compute the multi-partially symmetric rank-one approximations of multi-partially symmetric tensors via neural networks. Finally, we define the restricted $M$-singular values and the corresponding restricted $M$-singular vectors of higher-order multi-partially symmetric tensors and design to compute them. Numerical results show that the neural network models are efficient.

Xuezhong Wang, Maolin Che & Yimin Wei. (2020). Best Rank-One Approximation of Fourth-Order Partially Symmetric Tensors by Neural Network. Numerical Mathematics: Theory, Methods and Applications. 11 (4). 673-700. doi:10.4208/nmtma.2018.s01
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