Some Variations of Riemannian manifold HMC, MALA, and Lagrangian Monte Carlo
Diffusions-based and Hamiltonian dynamics-based methods, such as the Metropolis adjusted Langevin algorithms (MALA) and Hamiltonian Monte Carlo (HMC) algorithms have emerged as powerful Metropolis-Hastings algorithms. We consider some variations of the Riemannian manifold Hamiltonian (RMHMC) and Lagrangian Monte Carlo (LMC) methods. In particular, we investigate the mixtures of the LMC and RMHMC transition kernels with the manifold MALA kernels. The resulting algorithms converge at a geometric rate under certain conditions. The algorithms are illustrated using several examples.