  We propose a Monte Carlo sampler from the reverse diffusion process. Unlike
the practice of diffusion models, where the intermediary updates -- the score
functions -- are learned with a neural network, we transform the score matching
problem into a mean estimation one. By estimating the means of the regularized
posterior distributions, we derive a novel Monte Carlo sampling algorithm
called reverse diffusion Monte Carlo (rdMC), which is distinct from the Markov
chain Monte Carlo (MCMC) methods. We determine the sample size from the error
tolerance and the properties of the posterior distribution to yield an
algorithm that can approximately sample the target distribution with any
desired accuracy. Additionally, we demonstrate and prove under suitable
conditions that sampling with rdMC can be significantly faster than that with
MCMC. For multi-modal target distributions such as those in Gaussian mixture
models, rdMC greatly improves over the Langevin-style MCMC sampling methods
both theoretically and in practice. The proposed rdMC method offers a new
perspective and solution beyond classical MCMC algorithms for the challenging
complex distributions.
