High-Order Robust Discrete-Time Neural Dynamics for Time-Varying Multilinear Tensor Equation With $\mathcal {M}$-Tensor
Mei Liu, Huanmei Wu, Yang Shi, Long Jin
Abstract
The existing discrete-time neural dynamics methods for solving the multilinear tensor equation (MTE) with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {M}$</tex-math></inline-formula> -tensor are all derived from the continuous-time one and depend on the Euler difference formula, which cannot be applied to essentially discrete problems and have low solution accuracy. Moreover, these methods all focus on static problems rather than time-varying ones, and thus may have unsatisfactory performance in applications with time-varying parameters. Additionally, most of these methods fail to handle the MTE with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {M}$</tex-math></inline-formula> -tensor under noisy conditions. To remedy these issues, a high-order robust discrete-time neural dynamics (HRDND) method with a directly discrete approach is proposed for solving the time-varying MTE (TMTE) with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {M}$</tex-math></inline-formula> -tensor in this article. Theoretical analyses on convergence and robustness are provided to prove that the proposed HRDND method is feasible and effective. Finally, simulative experiments on four time-varying numerical examples and an application derived from the Bellman equation solved by the proposed HRDND method and other four methods are given, whose results illustrate the superiority of the proposed HRDND method.