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：正交矩阵在概率论与统计学中扮演着重要角色，尤其在高维统计模型中。使用正交矩阵对这类模型进行参数化有助于实现降维和参数识别。然而，从频率学派的视角为这些模型中的统计推断建立理论有效性颇具挑战，这促使研究者更倾向于采用贝叶斯方法，因其能够提供一致的不确定性量化。马尔可夫链蒙特卡洛方法通常用于后验分布的数值近似，而在由正交矩阵构成的Stiefel流形上进行采样存在显著困难。尽管已有多种策略被提出用于此目的，但基于参数化的梯度马尔可夫链蒙特卡洛方法效率最高。然而，现有文献缺乏对这些参数化方法的全面比较。本研究旨在填补这一空白，在同等条件下评估四种正交矩阵参数化替代方案的数值效率。评估针对四个问题展开。结果表明，极坐标展开参数化效率最高，尤其适用于高维复杂问题。然而，所有参数化方法在维度极高或难度较大的任务中均表现出局限性，这凸显了进一步发展正交矩阵采样方法的必要性。