Diffusion Posterior SMC Sampler

• Diffusion Posterior SMC Sampler is a framework for Bayesian inverse problems that integrates pretrained diffusion score models with sequential Monte Carlo for exact posterior inference.
• It initializes from a tractable prior and evolves particles via a modified Fokker–Planck PDE to rigorously approximate high-dimensional posteriors without heuristic approximations.
• Empirical validations in imaging tasks demonstrate its state-of-the-art reconstruction accuracy and scalability, supported by theoretical error bounds and convergence guarantees.

A Diffusion Posterior SMC (Sequential Monte Carlo) Sampler is a computational framework that enables statistically principled posterior inference for high-dimensional Bayesian inverse problems in which the prior is defined implicitly by a generative diffusion model. This approach avoids reliance on heuristic guidance approximations and instead rigorously evolves the posterior distribution using an ensemble of interacting particles governed by a modified Fokker–Planck (PDE) law. By leveraging pre-trained diffusion score models and simulating the evolution of the posterior via weighted SMC, it enables accurate, scalable, and asymptotically exact conditional sampling in settings such as computational imaging and scientific inverse problems.

1. Ensemble-Based Approximation-Free Posterior Evolution

The core of this methodology is an algorithm that marries the time-evolving structure of score-based diffusion models with the statistical rigor of SMC. The sampler operates in two stages:

1. Initialization: Particles are drawn from the unnormalized posterior at diffusion time $t=0$,

$\widehat{q}_y(\mathbf{x},0) \propto \widehat{\cev{p}_0}(\mathbf{x}) e^{-\mu_y(\mathbf{x})}$

where $\widehat{\cev{p}_0}$ is a tractable initial prior (commonly standard Gaussian), and $\mu_y(\mathbf{x}) = -\log p(y|\mathbf{x})$ encodes the negative log-likelihood.

1. Evolution: The particle ensemble $\{(\mathbf{x}^{(i)}_t,\, \beta^{(i)}_t)\}_{i=1}^N$ is moved forward in a simulated time parameter $t$ by integrating a modified partial differential equation (PDE) that exactly characterizes the evolution of the (conditional) posterior under diffusion. The key PDE is

$$\frac{\partial}{\partial t}\widehat{q}_y = -\nabla_{\mathbf{x}} \cdot \left((\widehat{b} - V^2 \nabla_{\mathbf{x}}\mu_y)\, \widehat{q}_y\right) + \frac{1}{2}V^2 \Delta_{\mathbf{x}} \widehat{q}_y + \left(U - \widehat{b}^\top \nabla_{\mathbf{x}}\mu_y - \mathbb{E}_{\widehat{q}_y}[U - \widehat{b}^\top \nabla_{\mathbf{x}}\mu_y]\right) \widehat{q}_y$$

where the drift $\widehat{b}(\mathbf{x},t)$ incorporates the pretrained score model, $V(t)$ is a diffusion schedule, $U$ is a second-order correction, and the final term ensures normalization.

This system is realized by an interacting, weighted SMC particle system: $$\begin{aligned} d\mathbf{x}_t^{(i)} &= (\widehat{b} - V^2 \nabla_{\mathbf{x}}\mu_y)(\mathbf{x}_t^{(i)},t)\, dt + V(t)\, d\mathbf{w}_t^{(i)} \ d\beta_t^{(i)} &= \Big[U(\mathbf{x}_t^{(i)}, t) - \widehat{b}(\mathbf{x}_t^{(i)}, t)^\top \nabla_{\mathbf{x}}\mu_y(\mathbf{x}_t^{(i)}) - \frac{1}{N} \sum_j \ldots \Big] \beta_t^{(i)}\, dt \end{aligned}$$ The weights $\beta_t^{(i)}$ correct for the evolving normalization and likelihood.

2. Diffusion Model as a Generative Prior

Diffusion models define a family of forward SDEs, gradually transitioning from tractable priors (often normal distributions) toward complex, learned target distributions. The pre-trained score function $\mathbf{s}_\theta(\mathbf{x}, t)$ approximates $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$ and underlies the backward-time SDE or ODE used for generation.

In the posterior setting, the drift and correction in the PDE are modified to steer the dynamics toward the true posterior distribution. The evolution thus dynamically incorporates both the learned data structure and the observed measurements through $\nabla_{\mathbf{x}}\mu_y$ (the likelihood gradient).

3. SMC Particle Methods and Reweighting

The key SMC innovation is the use of a weighted particle ensemble to represent the evolving posterior. Each particle carries a position and a weight,

• Particle positions are advanced under the diffusion (score-driven) drift, the likelihood guidance, and Brownian noise;
• Particle weights are updated continuously to absorb second-order corrections and to keep track of the normalizing constant.

Particles are occasionally resampled based on the effective sample size (ESS), ensuring the ensemble remains representative even in the face of degeneracy from sharply peaked likelihoods or ill-posed problems.

A corrector step (e.g., Langevin or ULA) can be included (ODE+corrector variant) to improve spread and robustness when the main PDE is integrated deterministically.

4. Error Bounds and Theoretical Guarantees

The approach is supported by explicit theoretical error analysis. The main theorem establishes that the error between the numerically simulated posterior $\widehat{q}_{y,T}$ and the ground-truth posterior $q_{y,0}$ is bounded as

$$TV(\widehat{q}_{y,T}, q_{y,0}) \leq C \sqrt{\frac{m_2^2}{T^2} + T^2 \epsilon^2}$$

where $m_2^2$ is a second-moment bound on the prior, $\epsilon^2$ is the $L^2$ score-matching training error, $T$ is the diffusion time, and $C$ is a problem-dependent constant. The error vanishes in the ideal limit of perfect score and infinite particles, and is optimized by balancing $T$.

Furthermore, as $N\to\infty$, the empirical distribution of the weighted ensemble converges in Wasserstein-2 to the target of the mean-field interacting particle system, thus establishing asymptotic exactness.

5. Empirical Validation in Imaging Inverse Problems

Empirical evaluation is carried out on large-scale, high-dimensional imaging inverse problems—such as Gaussian/motion deblurring, super-resolution, and inpainting—using established datasets like FFHQ-256 and ImageNet-256. The proposed AFDPS-SDE and AFDPS-ODE methods are benchmarked against state-of-the-art baselines, including DPS (heuristic guidance), SMC-FK-corrector variants, and split Gibbs samplers.

Results demonstrate that AFDPS matches or exceeds baselines in PSNR and perceptual similarity (LPIPS), and delivers more faithful, sharper reconstructions, particularly under strong corruption or adverse observation scenarios.

6. Practical Considerations and Applicability

The method is compatible with both SDE and ODE diffusion models, can incorporate pretrained score functions from large generative models, and scales to high-dimensional problems. Numerical integration can exploit parallel hardware, and the approach can be adapted for different inverse problem types and observation models.

Applications include, but are not limited to: computational imaging, physics, scientific inference, and any setting that can benefit from expressive, learned diffusion model priors for Bayesian inference.

7. Implications and Theoretical Significance

This methodology establishes a rigorous, principled foundation for posterior inference with diffusion priors. It overcomes limitations of heuristic or biased guidance approximations by deriving the exact posterior evolution from first principles. The error bounds link the quality of posterior inference directly to score model performance, and the SMC particle formulation provides a robust, scalable computational mechanism.

This formalism opens avenues for further research in interacting particle systems for conditional generative modeling, scalable Bayesian sampling with deep priors, and improved algorithms for high-dimensional, ill-posed inverse problems.

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Component	Description	Empirical Impact
Prior	Diffusion model, evolves via pre-trained score	Highly expressive, captures complex structures
Posterior evolution	PDE-derived, incorporates guidance and reweighting terms	Improved accuracy, asymptotic exactness
Computational implementation	Weighted SMC particle system (ensemble method)	Robust, scalable, parallelizable
Error control	Bound via training and ensemble size	Quantifiable and improvable
Empirical benchmarks	Challenging imaging inverse problems	State-of-the-art reconstructions