  Riemannian manifold Hamiltonian (RMHMC) and Lagrangian Monte Carlo (LMC) have
emerged as powerful methods of Bayesian inference. Unlike Euclidean Hamiltonian
Monte Carlo (EHMC) and the Metropolis-adjusted Langevin algorithm (MALA), the
geometric ergodicity of these Riemannian algorithms has not been extensively
studied. On the other hand, the manifold Metropolis-adjusted Langevin algorithm
(MMALA) has recently been shown to exhibit geometric ergodicity under certain
conditions. This work investigates the mixture of the LMC and RMHMC transition
kernels with MMALA in order to equip the resulting method with an "inherited"
geometric ergodicity theory. We motivate this mixture kernel based on an
analogy between single-step HMC and MALA. We then proceed to evaluate the
original and modified transition kernels on several benchmark Bayesian
inference tasks.
