Convergence of Kinetic Langevin Monte Carlo on Lie groups
Lingkai Kong Molei Tao
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.