This paper develops two novel and fast Riemannian second-order approaches for solving a class of matrix trace minimization problems with orthogonality constraints, which is widely applied in multivariate statistical analysis. The existing majorization method is guaranteed to converge but its convergence rate is at best linear. A hybrid Riemannian Newton-type algorithm with both global and quadratic convergence is proposed firstly. A Riemannian trust-region method based on the proposed Newton method is further provided. Some numerical tests and application to the least squares fitting of the DEDICOM model and the orthonormal INDSCAL model are given to demonstrate the efficiency of the proposed methods. Comparisons with some latest Riemannian gradient-type methods and some existing Riemannian second-order algorithms in the MATLAB toolbox Manopt are also presented.

中文翻译：

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解决多元统计中轨迹最小化问题的有效算法

本文提出了两种新颖且快速的黎曼二阶方法来解决一类具有正交性约束的矩阵迹线最小化问题，该方法已广泛应用于多元统计分析中。现有的主要化方法可以保证收敛，但是其收敛速度最好是线性的。首先提出了具有全局和二次收敛性的混合黎曼牛顿型算法。进一步提出了一种基于牛顿法的黎曼信赖域方法。给出了一些数值测试并将其应用于DEDICOM模型和正交INDSCAL模型的最小二乘拟合，以证明所提出方法的有效性。