  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.
