Orthogonal matrices play an important role in probability and statistics, particularly in high-dimensional statistical models. Parameterizing these models using orthogonal matrices facilitates dimension reduction and parameter identification. However, establishing the theoretical validity of statistical inference in these models from a frequentist perspective is challenging, leading to a preference for Bayesian approaches because of their ability to offer consistent uncertainty quantification. Markov chain Monte Carlo methods are commonly used for numerical approximation of posterior distributions, and sampling on the Stiefel manifold, which comprises orthogonal matrices, poses significant difficulties. While various strategies have been proposed for this purpose, gradient-based Markov chain Monte Carlo with parameterizations is the most efficient. However, a comprehensive comparison of these parameterizations is lacking in the existing literature. This study aims to address this gap by evaluating numerical efficiency of the four alternative parameterizations of orthogonal matrices under equivalent conditions. The evaluation was conducted for four problems. The results suggest that polar expansion parameterization is the most efficient, particularly for the high-dimensional and complex problems. However, all parameterizations exhibit limitations in significantly high-dimensional or difficult tasks, emphasizing the need for further advancements in sampling methods for orthogonal matrices.

翻译：正交矩阵在概率与统计学中扮演重要角色，特别是在高维统计模型中。使用正交矩阵对这些模型进行参数化有助于降维和参数识别。然而，从频率学派角度建立这些模型统计推断的理论有效性颇具挑战性，因此贝叶斯方法因其能够提供一致的不确定性量化而更受青睐。马尔可夫链蒙特卡洛方法通常用于后验分布的数值逼近，而在由正交矩阵构成的Stiefel流形上采样面临显著困难。尽管已有多种策略被提出用于此目的，但基于梯度的参数化马尔可夫链蒙特卡洛方法最为高效。然而，现有文献缺乏对这些参数化方法的全面比较。本研究旨在通过评估四种正交矩阵替代参数化方法在同等条件下的数值效率来填补这一空白。评估在四个问题场景中进行。结果表明，极坐标展开参数化方法最为高效，尤其适用于高维和复杂问题。然而，所有参数化方法在极高维度或困难任务中均表现出局限性，这凸显了正交矩阵采样方法需要进一步改进的必要性。