It is well-known that irreversible MCMC algorithms converge faster to their stationary distributions than reversible ones. Using the special geometric structure of Lie groups $\mathcal G$ and dissipation fields compatible with the symplectic structure, we construct an irreversible HMC-like MCMC algorithm on $\mathcal G$, where we first update the momentum by solving an OU process on the corresponding Lie algebra $\mathfrak g$, and then approximate the Hamiltonian system on $\mathcal G \times \mathfrak g$ with a reversible symplectic integrator followed by a Metropolis-Hastings correction step. In particular, when the OU process is simulated over sufficiently long times, we recover HMC as a special case. We illustrate this algorithm numerically using the example $\mathcal G = SO(3)$.
翻译:众所周知,不可逆的MCMC算法比可逆的算法更快地聚集到其固定的分布上。我们利用Lie组的特殊几何结构 $\ mathcal G$ 和与共振结构兼容的分散字段,用$\ mathcal G$ 构建一个不可逆的HMC 类似MC MC 算法,我们首先通过在相应的Lie algebra $\ mathfrak g$上解决OU进程来更新势头,然后以 $mathcal G\ times\ mathfrak g$ 来接近汉密尔顿系统。 我们用 $\ mathcal G = SO(3)$ 来用可逆的共振集器和大都会-Hastings 校正步骤来补充这一算法。 特别是当OU进程被模拟了足够长的时间,我们用一个特例来恢复HMC。 我们用以 $\ mathcal G = SO(3)$来用数字说明这一算法。