Neural networks for computing best rank-one approximations of tensors and its applications

This paper presents the neural dynamical network to compute a best rank-one approximation of a real-valued tensor. We implement the neural network model by the ordinary differential equations (ODE), which is a class of continuous-time recurrent neural network. Several new properties of solutions for the neural network are established. We prove that the locally asymptotic stability of solutions for ODE by constructive an appropriate Lyapunov function under mild conditions. Furthermore, we also discuss how to use the proposed neural networks for solving the tensor eigenvalue problem including the tensor H-eigenvalue problem, the tensor Z-eigenvalue problem, and the generalized eigenvalue problem with symmetric-definite tensor pairs. Finally, we generalize the proposed neural networks to the computation of the restricted singular values and the associated restricted singular vectors of real-valued tensors. We illustrate and validate theoretical results via numerical simulations.