Reverse Diffusion Monte Carlo
This provides a new solution for sampling from challenging complex distributions, potentially impacting fields like machine learning and statistics, though it appears incremental as an alternative to existing MCMC methods.
The authors tackled the problem of sampling from complex distributions by proposing a reverse diffusion Monte Carlo (rdMC) method, which transforms score matching into mean estimation and can sample with any desired accuracy, demonstrating significantly faster sampling than MCMC for multi-modal distributions like Gaussian mixture models.
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.