Stiefel流形的极展开蒙特卡罗模拟
Monte Carlo simulation on the Stiefel manifold via polar expansion
摘要：
出于贝叶斯推理的应用程序与正交矩阵参数的统计模型，我们提出$\textit{polar expansion},$的概率分布的Stiefel流形上的蒙特卡罗模拟的一般方法。为了绕过从随机正交矩阵$\boldsymbol{Q},$的分布进行模拟的许多公认的挑战，我们构造无约束随机矩阵$\boldsymbol{X}$的分布，使得$\boldsymbol{X},$的极分解的正交分量$\boldsymbol{Q}_X,$在分布上等于$\boldsymbol{Q}.$。$\boldsymbol{X}$的分布服从使用标准方法的马尔可夫链蒙特卡罗（MCMC）模拟，并且可以从无约束空间上的马尔可夫链恢复$\boldsymbol{Q}$的分布的近似。当与现代MCMC软件相结合时，极坐标展开允许在具有正交矩阵参数的模型中进行常规和灵活的后验推断。我们发现，极性扩展与自适应哈密顿蒙特卡罗是一个数量级更有效的竞争MCMC方法在一个基准的蛋白质相互作用网络的应用。我们还提出了一种新的方法，贝叶斯函数主成分分析，我们说明在气象时间序列应用。
英文摘要：
Motivated by applications to Bayesian inference for statistical models with orthogonal matrix parameters, we present $\textit{polar expansion},$ a general approach to Monte Carlo simulation from probability distributions on the Stiefel manifold. To bypass many of the well-established challenges of simulating from the distribution of a random orthogonal matrix $\boldsymbol{Q},$ we construct a distribution for an unconstrained random matrix $\boldsymbol{X}$ such that $\boldsymbol{Q}_X,$ the orthogonal component of the polar decomposition of $\boldsymbol{X},$ is equal in distribution to $\boldsymbol{Q}.$ The distribution of $\boldsymbol{X}$ is amenable to Markov chain Monte Carlo (MCMC) simulation using standard methods, and an approximation to the distribution of $\boldsymbol{Q}$ can be recovered from a Markov chain on the unconstrained space. When combined with modern MCMC software, polar expansion allows for routine and flexible posterior inference in models with orthogonal matrix parameters. We find that polar expansion with adaptive Hamiltonian Monte Carlo is an order of magnitude more efficient than competing MCMC approaches in a benchmark protein interaction network application. We also propose a new approach to Bayesian functional principal components analysis which we illustrate in a meteorological time series application.
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