  Probability measures supported on submanifolds can be sampled by adding an
extra momentum variable to the state of the system, and discretizing the
associated Hamiltonian dynamics with some stochastic perturbation in the extra
variable. In order to avoid biases in the invariant probability measures
sampled by discretizations of these stochastically perturbed Hamiltonian
dynamics, a Metropolis rejection procedure can be considered. The so-obtained
scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC)
algorithms. We show here how to generalize to GHMC a procedure suggested by
Goodman, Holmes-Cerfon and Zappa for Metropolis random walks on submanifolds,
where a reverse projection check is performed to enforce the reversibility of
the algorithm for large timesteps and hence avoid biases in the invariant
measure. We also provide a full mathematical analysis of such procedures, as
well as numerical experiments demonstrating the importance of the reverse
projection check on simple toy examples.
