Exact Global MCMC with Denoising Diffusion

This work shows that diffusion models learned with the standard denoising loss provide exact and effective global MCMC proposals for complex high-dimensional target densities. A forward noising process followed by its exact reversal is a Markov chain that leaves the target invariant, and a denoising diffusion model is a learned approximation of that reversal. A Metropolis–Hastings (MH) step whose acceptance ratio includes the forward and reverse path densities of a discrete-time SDE makes the chain exact for any denoiser, with an acceptance rate that measures how closely the denoiser approximates the ideal reversal. We train denoising diffusion models on locally convergent samples from the Metropolis-adjusted Langevin algorithm (MALA), whose weighting across modes need not match the target since the MH step absorbs the mismatch, and call the composition of the resulting global path sampler with a local MALA sampler Denoising Diffusion Monte Carlo (DDMC). Across Gaussian mixtures, particle systems, and a 550-dimensional Bayesian neural network posterior, DDMC matches ground-truth energy distributions at the measured estimator floor and recovers correct mode weights. On the LJ-38 double-funnel potential below its melting temperature, DDMC obtains the equilibrium funnel occupancy from a mis-weighted training dataset, agrees with parallel tempering, and attains an order of magnitude lower variance at equal total compute, dataset generation and denoiser training included.