Explicit, momentum-based dynamics for optimizing functions defined on Lie groups was recently constructed, based on techniques such as variational optimization and left trivialization. We appropriately add tractable noise to the optimization dynamics to turn it into a sampling dynamics, leveraging the advantageous feature that the trivialized momentum variable is Euclidean despite that the potential function lives on a manifold. We then propose a Lie-group MCMC sampler, by delicately discretizing the resulting kinetic-Langevin-type sampling dynamics. The Lie group structure is exactly preserved by this discretization. Exponential convergence with explicit convergence rate for both the continuous dynamics and the discrete sampler are then proved under $W_2$ distance. Only compactness of the Lie group and geodesically $L$-smoothness of the potential function are needed. To the best of our knowledge, this is the first convergence result for kinetic Langevin on curved spaces, and also the first quantitative result that requires no convexity or, at least not explicitly, any common relaxation such as isoperimetry.

翻译：基于变分优化和左平凡化等技术，最近构建了用于优化李群上定义函数的显式、基于动量的动力学。我们通过向优化动力学中添加易于处理的噪声，将其转化为采样动力学，这得益于一个有利特性：尽管势函数定义在流形上，平凡化后的动量变量却是欧几里得空间的。随后，我们通过精细离散化所得的动力学朗之万型采样动力学，提出了一种李群MCMC采样器。该离散化过程严格保持了李群结构。我们在$W_2$距离下证明了连续动力学和离散采样器均具有显式收敛率的指数收敛性。仅需李群的紧致性及势函数的测地$L$-光滑性。据我们所知，这是弯曲空间上动力学朗之万方法的首个收敛性结果，也是首个无需凸性条件或（至少不明显依赖）等周不等式等常见松弛条件的定量结果。