Particle Denoising Diffusion Sampler

TL;DR

PDDS offers a principled way to sample from unnormalized densities by uniting guided denoising diffusion with a sequential Monte Carlo backbone. The method introduces a novel score-matching loss to learn guiding potentials, and provides theoretical guarantees for finite-particle consistency and asymptotic behavior under discretization, including results for both fixed-step and infinitely fine discretizations. Learning-based potentials via NSM mitigate variance and improve target fidelity, enabling iterative refinements that enhance normalizing-constant estimation and sample quality. Empirical results across multimodal and high-dimensional tasks demonstrate PDDS’s competitive performance against strong baselines, with added flexibility to incorporate MCMC steps for further gains.

Abstract

Denoising diffusion models have become ubiquitous for generative modeling. The core idea is to transport the data distribution to a Gaussian by using a diffusion. Approximate samples from the data distribution are then obtained by estimating the time-reversal of this diffusion using score matching ideas. We follow here a similar strategy to sample from unnormalized probability densities and compute their normalizing constants. However, the time-reversed diffusion is here simulated by using an original iterative particle scheme relying on a novel score matching loss. Contrary to standard denoising diffusion models, the resulting Particle Denoising Diffusion Sampler (PDDS) provides asymptotically consistent estimates under mild assumptions. We demonstrate PDDS on multimodal and high dimensional sampling tasks.

Figures (20)

• Figure 1: Tempered (top) and noised (bottom) sequences of distributions for the target $\pi(x) = 0.8 \mathcal{N}(x; -4, 0.5^2) + 0.2 \mathcal{N}(x; 4, 1)$. The tempered sequence follows $\pi_t(x) \propto \pi(x)^{(1-\eta_t)} \phi(x)^{\eta_t}$ where $\phi$ is the standard normal and $\eta_t$ increases from $0$ to $1$. The noising sequence follows the forward diffusion in \ref{['eq:forward_diffusion']}. Red dots indicate the position of modes in the original target. The tempered sequence suffers from mode switching, i.e. the low mass large width mode becomes dominant across the tempered path. The noised sequence does not suffer from this problem.
• Figure 2: $\log \hat{\mathcal{Z}}^N_0$ for our method (PDDS and PDDS-MCMC), compared with SMC, CRAFT, DDS and PIS. Dotted black represents analytic ground truth where available, otherwise long-run SMC. Variation is displayed over both training and sampling seeds (2000 total). The y-axes on Sonar and LGCP have been cropped and outliers (present in all methods) removed for clarity. Uncurated samples are presented in \ref{['app:additional_results']}.
• Figure 3: $\log \hat{\mathcal{Z}}^N_0$ for CRAFT and PDDS-MCMC on the GMM task in 20 dimensions. Variation displayed over training and sampling seeds (1000 total).
• Figure 4: Samples from each method on the Gaussian Mixture task with 4 steps.
• Figure 5: ESS curves on the Sonar task with 200 steps. Left: PDDS with approximation \ref{['eq:naive_approximation']}, right: PDDS with learnt potential approximation.

Theorems & Definitions (26)

• Lemma 2.1
• Proposition 2.2
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 4.1
• Proposition 4.2
• Lemma 2.1
• Lemma 2.2
• Definition 2.3