Accurate Sampling from Diffusion Models

Dénes Sexty

hep-latSep 24, 2026

Accurate Sampling from Diffusion Models

Authors: Dénes Sexty

Organizations: Institute of Physics, NAWI Graz, University of Graz, Universitätsplatz 5, 8010 Graz, Austria

Abstract

A new proposal called DM-SMC (Diffusion Model - Sequential Monte Carlo) is investigated, which samples ensembles defined in terms of an action, using diffusion models trained on samples from the ensemble. The SMC setup allows for accurate sampling in spite of an approximate diffusion model and the finite stepsize used in the numerical solution of the stochastic process. Improved update strategies are also investigated. Results are presented for a $Z_2$ symmetric scalar field theory in 2 dimensions near its 2nd order phase transition.

Figures & tables

Panel of Figure 1 — Figure 1: The effective sample size over the number of total samples on a 4x4 lattice for $\kappa=0.27$ and $\lambda=0.022$ and various Langevin stepsizes, as indicated by the total number of steps. The first order Euler-Maruyama update is used. On the left, the learning of the model was stopped 200 learning steps. On the right, the learning was allowed to converge using $\approx 1000$ learning steps, and the parameters of the model were averaged for a further 1200 steps. No resampling was used.

Figure 2: Histogram of the field average $\Phi$ for a 4x4 lattice for 2d scalar field theory using $\kappa=0.27$ and $\lambda=0.022$ . Configurations are sampled using denoising (without SMC setup) with the two models in Fig. 1 and $N_{\tau}=200$ steps for the “bad” model and $N_{\tau}=200$ steps with improved update steps for the “good” model. Results from an HMC simulation are shown as well.

Panel of Figure 3 — Figure 3: Histogram of the field average $\Phi$ (left panel) and histogram for the action density $s$ (right panel) for a 4x4 lattice for 2d scalar field theory using $\kappa=0.27$ and $\lambda=0.022$ . Measurements using the DM-SMC setup use the model corresponding to the left panel of Fig. 1 , $N_{\tau}=200$ , and one resampling at the end of the denoising process. Results from an HMC simulation are shown as well.

Panel of Figure 4 — Figure 4: The ESS is shown for an improved denoising update for various Langevin stepsizes, as indicated. In the left panel, the determinant of the Jacobian is calculated exactky. In the right panel, the denterminant of the Jacobian is approximated using 4 noise vectors with the formulas in eqs. ( 25 ) and ( 26 ).

Figure 5: The average of $\phi^{2}$ is shown for various stepsizes, as calculated from DM-SMC using the 2nd order update and the approximation of the Jacobian with 4 noise vectors. Extrapolation to zero stepsize, using stepsizes $\leq 0.01$ , and an HMC benchmark value is also shown.

Panel of Figure 6 — Figure 6: Histogram of the field average $\Phi$ (left panel) and histogram for the action density $s$ (right panel) for a $16^{2}$ lattice for 2d scalar field theory using $\kappa=0.27$ and $\lambda=0.022$ . The diffusion process uses $\sigma=50$ . Denoising is done using $N_{\tau}=1000$ , and systematic resampling when the ESS/ $M$ decreases below 0.5. Results from an HMC simulation are shown as well.

Explore similar work

Exact Global MCMC with Denoising Diffusion
Aug 31, 2026Mitch HillMarkov Chain Monte Carlo Markov Models
Non-Asymptotic Error Bounds for SMC with Biased Proposals: Application to Conditional Diffusion Sampling
Jul 6, 2026Stanislas Strasman, Gabriel Victorino Cardoso, Sylvain Le Corff +2Markov Chain Monte Carlo Markov Models
SU, LPSM · Sorbonne Université and Université Paris Cité, CNRS, LPSM, F-75005 Paris, France · LPSM +2
Diffusion model for SU(N) gauge theories
May 7, 2026Javad Komijani, Marina K. Marinkovic, Lara TurgutScore Matching Diffusion Sampling
Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

Aug 31, 2026J/K move · Enter open · S save

Exact Global MCMC with Denoising Diffusion

Mitch Hill

Markov Chain Monte Carlo Markov Models Diffusion Models

Open paper AskPDF arXiv

This work shows that diffusion models learned with standard denoising loss can provide effective global MCMC proposals for complex high-dimensional target densities. The method is motivated by the observation that sequentially applying a forward and reverse diffusion process defines a Markov chain with a target stationary distribution for an ideal denoiser trained on samples of the target distribution. This observation can be made exact for any denoiser by applying a Metropolis-Hastings step whose acceptance ratio includes the density of the forward and reverse paths of a discrete time SDE approximation. We therefore propose to train denoising diffusion models on locally convergent MALA samples to learn global MCMC proposals. We call the composition of the global denoiser-based path sampler and a local MALA sampler Denoising Diffusion Monte Carlo (DDMC). Experiments show that DDMC can provide global proposals with high acceptance across a variety of complex target densities. Our results offer preliminary evidence that the established scaling behavior of standard diffusion training transfers directly to exact sampling from high-dimensional unnormalized densities.

Abstract

Explore similar work

Exact Global MCMC with Denoising Diffusion

Non-Asymptotic Error Bounds for SMC with Biased Proposals: Application to Conditional Diffusion Sampling

Diffusion model for SU(N) gauge theories