The adaptive cubic regularization algorithm employing the inexact gradient and Hessian is proposed on general Riemannian manifolds, together with the iteration complexity to get an approximate second-order optimality under certain assumptions on accuracies about the inexact gradient and Hessian. The algorithm extends the inexact adaptive cubic regularization algorithm under true gradient in [Math. Program., 184(1-2): 35-70, 2020] to more general cases even in Euclidean settings. As an application, the algorithm is applied to solve the joint diagonalization problem on the Stiefel manifold. Numerical experiments illustrate that the algorithm performs better than the inexact trust-region algorithm in [Advances of the neural information processing systems, 31, 2018].

翻译：本文在一般黎曼流形上提出了采用不精确梯度和Hessian的自适应三次正则化算法，并在关于不精确梯度和Hessian精度的特定假设下，给出了达到近似二阶最优性的迭代复杂度。该算法将[Math. Program., 184(1-2): 35-70, 2020]中基于精确梯度的不精确自适应三次正则化算法推广至更一般的情形，即使在欧几里得设定下也是如此。作为应用，该算法被用于求解Stiefel流形上的联合对角化问题。数值实验表明，该算法性能优于[Advances of the neural information processing systems, 31, 2018]中的不精确信赖域算法。