Parameterizations for Gradient-based Markov Chain Monte Carlo on the Stiefel Manifold: A Comparative Study

Abstract: 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.