Best Rank-One Approximation of Fourth-Order Partially Symmetric Tensors by Neural Network

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.