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 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 W2 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：近期，基于变分优化与左平凡化等技术，研究者构建了用于优化李群上函数的显式动量动力学。我们通过向优化动力学中恰当添加可处理噪声，将其转化为采样动力学，并利用动量变量为欧几里得向量而势函数定义在流形上的有利特性。随后，通过精细离散化所得到的动力学朗之万型采样动力学，我们提出了一种李群马尔可夫链蒙特卡洛采样器。该离散化过程精确保持了李群结构。我们证明了连续动力学与离散采样器在W2距离下均具有显式收敛率的指数收敛性，且仅需李群的紧致性与势函数的测地L-光滑性条件。据我们所知，这是弯曲空间上动力学朗之万的首个收敛性结果，也是首个无需凸性假设或任何常见松弛条件（如等周不等式）的定量收敛性结果。