Numerically efficient and stable algorithms are essential for kernel-based regularized system identification. The state of art algorithms exploit the semiseparable structure of the kernel and are based on the generator representation of the kernel matrix. However, as will be shown from both the theory and the practice, the algorithms based on the generator representation are sometimes numerically unstable, which limits their application in practice. This paper aims to address this issue by deriving and exploiting an alternative Givens-vector representation of some widely used kernel matrices. Based on the Givens-vector representation, we derive algorithms that yield more accurate results than existing algorithms without sacrificing efficiency. We demonstrate their usage for the kernel-based regularized system identification. Monte Carlo simulations show that the proposed algorithms admit the same order of computational complexity as the state-of-the-art ones based on generator representation, but without issues with numerical stability.

翻译：数值高效且稳定的算法对于基于核的正则化系统辨识至关重要。现有先进算法利用核的半可分结构，并基于核矩阵的生成元表示。然而，理论与实际应用均表明，基于生成元表示的算法有时存在数值不稳定性，这限制了其实际应用。本文旨在通过推导并利用若干广泛使用的核矩阵的替代性Givens向量表示来解决此问题。基于Givens向量表示，我们推导出在保持效率的同时比现有算法精度更高的算法，并演示其在基于核的正则化系统辨识中的应用。蒙特卡洛仿真表明，所提算法与基于生成元表示的先进算法具有相同阶数的计算复杂度，且无数值稳定性问题。