  We propose a solution for linear inverse problems based on higher-order
Langevin diffusion. More precisely, we propose pre-conditioned second-order and
third-order Langevin dynamics that provably sample from the posterior
distribution of our unknown variables of interest while being computationally
more efficient than their first-order counterpart and the non-conditioned
versions of both dynamics. Moreover, we prove that both pre-conditioned
dynamics are well-defined and have the same unique invariant distributions as
the non-conditioned cases. We also incorporate an annealing procedure that has
the double benefit of further accelerating the convergence of the algorithm and
allowing us to accommodate the case where the unknown variables are discrete.
Numerical experiments in two different tasks in communications (MIMO symbol
detection and channel estimation) and in three tasks for images showcase the
generality of our method and illustrate the high performance achieved relative
to competing approaches (including learning-based ones) while having comparable
or lower computational complexity.
