Robust Diffusion Posterior Sampling

• The paper presents robust diffusion posterior sampling, integrating adaptive guidance and generalized Bayesian losses to ensure uniform influence bounds under measurement perturbations.
• It employs innovative adaptive likelihood scaling and restart strategies that improve error contraction and sample quality across diverse inverse problems.
• The methodology enhances consistency, uncertainty quantification, and provable convergence guarantees, broadening its applicability in imaging and scientific data recovery.

Robust Diffusion Posterior Sampling is a principled framework for sampling from Bayesian posteriors using diffusion models as expressive priors, with algorithmic modifications and architectural innovations designed to guarantee stability, statistical robustness, and operational resilience under model misspecification and adverse noise conditions. The robustness of a diffusion posterior sampler encompasses its ability to limit the impact of deviations in the assumed observation model and data-distributional shifts, while maintaining consistency, sample quality, and uncertainty quantification across a diverse set of inverse problems. Recent advances embed robustness via adaptive guidance strategies, generalized Bayesian loss functions, coupled stochastic dynamics, or plug-and-play correction, endowing these methods with provable and empirical resilience to outliers, heavy-tailed measurement noise, and operator mismatches (Yang et al., 2 Feb 2026, Hen et al., 23 Nov 2025, &&&2&&&).

1. Background: Diffusion Models for Bayesian Inverse Problems

Diffusion models define expressive priors $p(x)$ over high-dimensional signals via a parameterized stochastic process—typically a Markov process or SDE/ODE—mapping samples from a base Gaussian distribution to a data-like manifold through learned “score” vectors $s_\theta(x, t) \approx \nabla_x \log p_t(x)$. The standard Bayesian inverse problem is to infer $x_0$ given noisy/incomplete measurements $y = F(x_0) + \epsilon$ with prior $p(x_0)$. Sampling from the posterior $p(x_0 | y)$ is generally intractable due to the nonlinearity and high dimensionality of both the prior and likelihood.

Diffusion posterior sampling (DPS) extends the learned reverse diffusion process to the posterior by augmenting the unconditional score with a likelihood-informed correction: $$\nabla_{x_t}\log p_t(x_t|y) = \nabla_{x_t}\log p_t(x_t) + \nabla_{x_t}\log p_t(y|x_t)$$ and approximating the intractable term $\nabla_{x_t}\log p_t(y|x_t)$ with a surrogate based on Tweedie’s formula and the measurement model (Chung et al., 2022, Syarubany, 25 Dec 2025). Although this achieves strong results, its robustness degrades sharply if the likelihood model deviates from the true data-generating process or if the measurements are corrupted by heavy-tailed noise or outliers (Yang et al., 2 Feb 2026).

2. Limitations of Standard Diffusion Posterior Sampling

Classical DPS and related plug-and-play solvers are provably stable in a local-Lipschitz sense via the Girsanov theorem and relative entropy analysis: the KL divergence between two posterior distributions is controlled by the difference in measurements (Yang et al., 2 Feb 2026): $$\mathrm{D}_{\mathrm{KL}}(\widetilde{p}(\cdot|y) \| \widetilde{p}(\cdot|y')) \leq C_{\mathrm{stable}} \|y - y'\|^2$$ However, these methods are not robust in the sense of uniformly bounding the Posterior Influence Function (PIF) under arbitrary measurement perturbations: $$\mathrm{PIF}_{y_*}(y) = \mathrm{D}_{\mathrm{KL}}(p(\cdot|y_*) \| p(\cdot|y))$$ This lack of robustness is evidenced by unbounded KL divergence as measurement perturbation grows, leading to catastrophic error propagation under model misspecification, heavy-tailed noise, or adversarial outliers (Yang et al., 2 Feb 2026). Empirical studies show that plug-and-play/diffusion posterior solvers fail on Student-t or impulsive noise, producing artifacts and sharp loss in recovery metrics (Yang et al., 2 Feb 2026).

3. Robustification Strategies

3.1 Generalized Bayesian Losses and Influence Bounds

Robust Diffusion Posterior Sampling (RDP) replaces the classical negative log-likelihood in the posterior update with a robust loss function $\ell_y(x)$ incorporating a smooth, bounded weighting function $w(r)$ for residuals: $$p^{(\ell)}(x|y) \propto p(x) \exp\{-\tau \ell_y(x)\},\quad \ell_y(x) = -\sum_{i} w(r_i) \log \tilde{p}(y_i | \hat{x}_{0|t})$$ with guidance: $$\hat{s} = s_\theta(x_t, t) - \tau_t \nabla_{x_t} \ell_y(x_t)$$ If the weight function satisfies $|r w(r)| < \infty$ and $|r^2 w'(r)| < \infty$ for all $r$, this construction provably bounds the posterior influence uniformly for all $y$ [(Yang et al., 2 Feb 2026) (Theorem 2)], guaranteeing that no measurement perturbation can cause divergence in the sampled posterior. In practice, choices like the IMQ-weight $w(r)=(1 + r^2/c^2)^{-1/2}$ enforce outlier rejection.

3.2 Adaptive Likelihood Guidance

Observation-dependent adaptation of the likelihood step-size can automatically balance the contribution of the measurement term along the reverse diffusion (Hen et al., 23 Nov 2025): $$x_{t-1} = \text{DDIM}(x_t) - 2 \gamma_t \frac{\langle d_t, g_t \rangle}{\|g_t\|_2^2} g_t$$ with $g_t$ the surrogate likelihood gradient, and $d_t$ a MAP-based residual. This alignment-based scaling dynamically shrinks the likelihood influence in regions where the prior and likelihood are misaligned, mitigating the risk of over-correction or artifacts (Hen et al., 23 Nov 2025).

3.3 Coupled Data and Measurement Space Dynamics

C-DPS introduces two tightly coupled forward diffusions in data and measurement space, so that at every reverse step, the recursion

$$p(x_{t-1}|x_t, y_{t-1}) \propto p(x_t|x_{t-1})\,p(y_{t-1}|x_{t-1})$$

yields a closed-form Gaussian posterior update that enforces measurement consistency exactly throughout the entire sampling trajectory—eliminating the need for heuristic projections or surrogate gradients and ensuring robust fidelity to the measurements even under severe noise or operator misspecification (Hamidi et al., 8 Oct 2025).

3.4 Restart/Ensemble Strategies and Plug-and-Play Mechanisms

Restart Posterior Sampling (RePS) interleaves short ODE-like sampling segments with noise reinjection, contracting accumulated model or discretization errors and decoupling measurement conditioning from score network backpropagation. This approach is robust to both local approximation and global model bias, especially in the presence of strong non-Gaussian effects (Ahmed et al., 24 Nov 2025). Similarly, plug-and-play frameworks (DPnP) rigorously alternate between exact data-fidelity steps and denoising-diffusion steps, providing non-asymptotic total variation and spectral gap guarantees for the overall chain, including nonlinear measurement operators (Xu et al., 2024).

3.5 Discrete and Ensemble Samplers

Split Gibbs Discrete Diffusion Posterior Sampling (SGDD) offers provable convergence to the discrete-state posterior by alternating between likelihood-guided Metropolis-Hastings and prior-guided discrete diffusion, with quantitative bounds on Fisher divergence and empirical improvements on DNA sequence design, discrete MNIST, and music infilling benchmarks (Chu et al., 3 Mar 2025). Approximation-free ensemble samplers simulate the continuous posterior Fokker-Planck PDE with weighted particle systems, ensuring theoretical error bounds in TV distance controlled by score error and number of particles (Chen et al., 4 Jun 2025).

4. Theoretical Guarantees and Empirical Validation

Robust diffusion posterior samplers admit new theoretical guarantees:

• Uniform influence bounds: For appropriately weighted robust losses, the KL-divergence between the correct and robustified posterior is uniformly bounded for all measurement perturbations, in contrast to the unbounded error of classical solvers (Yang et al., 2 Feb 2026).
• Awareness-adaptive step size: Adaptive guidance with alignment-based weighting auto-calibrates the likelihood contribution, allowing for provable resilience to operator re-spacing, stochasticity, and increasing observation noise (Hen et al., 23 Nov 2025).
• Non-asymptotic mixing and TV error: Plug-and-play schemes (DPnP) provably mix to the perturbed posterior within O(1/(1-λ) log(1/ε)) steps, with a spectral gap λ bounded away from one, and errors scaling with per-step MALA and denoising inaccuracies (Xu et al., 2024).
• Empirical improvement under misspecification: RDP and related robust schemes yield up to 20–50% improvements in LPIPS, PSNR, and FID over baselines on phase retrieval, deblurring, inpainting, and outlier-contaminated scientific imaging, without loss under well-specified Gaussian likelihoods (Yang et al., 2 Feb 2026, Hen et al., 23 Nov 2025, Hamidi et al., 8 Oct 2025).

5. Algorithmic and Practical Implications

The following summarizes robust algorithmic design principles and practical deployment:

• Loss engineering: Choice of robust loss/weight for the likelihood term is essential for outlier resilience; IMQ/fair/huber or similar functions can be plugged into any guided diffusion sampler (Yang et al., 2 Feb 2026).
• Adaptive schedules: Measurement-aware or alignment-based scaling of the likelihood gradient can entirely eliminate the need for manual tuning of guidance strength or pre-schedule annealing (Hen et al., 23 Nov 2025).
• Coupled forward diffusions: Synchronizing data and measurement noise processes enables exact Bayesian recursions at every step and prevents drift even in the presence of misestimation or heavy-handed measurement errors (Hamidi et al., 8 Oct 2025).
• Restart and plug-and-play: Periodic restarts or hybrid proximal-diffusion alternations ensure error contraction and can handle both linear and highly nonlinear or nonconvex forward operators (Ahmed et al., 24 Nov 2025, Xu et al., 2024).
• Universality: Robust strategies are compatible with continuous and discrete diffusion models (SGDD), as well as with approximating samplers (ensemble-based, message passing) in high-dimensional Bayesian statistics (Chu et al., 3 Mar 2025, Montanari et al., 2023, Chen et al., 4 Jun 2025).
• Computational cost: Additional cost arises from evaluating robust weighted likelihoods, solving MAP or CG subproblems (for ensemble or coupled samplers), or adaptively scaling gradients, but is offset by gains in sampling efficiency, stability, and robustness.

6. Applications, Limitations, and Future Directions

Robust diffusion posterior sampling frameworks have been successfully deployed in:

• Medical imaging: Accelerated MRI with preconditioned ULA for rapid, reliable posterior sampling and calibrated uncertainty—without need for task-specific hyperparameter tuning (Blumenthal et al., 5 Dec 2025).
• Noisy nonlinear inverse problems: Deblurring, inpainting, super-resolution, and phase retrieval, as well as scientific inverse scattering, under heavy-tailed and adversarial measurement errors (Yang et al., 2 Feb 2026, Hamidi et al., 8 Oct 2025, Zhou et al., 2024).
• Discrete combinatorial domains: DNA design, discrete image recovery, and symbolic music infilling via discrete-state split-Gibbs samplers (Chu et al., 3 Mar 2025).

Notable limitations remain: hyperparameter tuning is still needed for certain robust loss parameters; computational cost increases for CG/MAP subroutines; and some strategies (e.g., latent-domain extension for AdaPS) are not yet fully realized (Hen et al., 23 Nov 2025). Open research directions include inference under unknown or partially observed forward operators, universal robustification with provable domain transfer, and cross-domain extensions to sequence, graph, and manifold-valued data.

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In summary, robust diffusion posterior sampling represents a convergence of algorithmic innovation and statistical rigor, providing principled tools for resilient Bayesian inference in high-dimensional and adverse settings. Current methodologies deliver a spectrum of robustness guarantees—spanning uniform influence bounds, adaptive likelihood scaling, and plug-and-play or ensemble convergence—and demonstrate broad empirical advances across imaging, scientific, and symbolic domains (Yang et al., 2 Feb 2026, Hen et al., 23 Nov 2025, Hamidi et al., 8 Oct 2025, Blumenthal et al., 5 Dec 2025, Chu et al., 3 Mar 2025).