Learning to Replace MCMC in Split-Gibbs Diffusion Posterior Sampling via Deep Unfolding

• The likelihood-side MCMC can require sample-dependent iteration counts, complicate parallelization, require tuning of stepsizes or proposal scales, and incur substantial computational cost.Source: §1 · p. 1
• The proposed learning-based framework replaces iterative likelihood-side MCMC while preserving modular prior-likelihood computation.Source: §1 · p. 1
• Both split-Gibbs updates are reformulated as Gaussian denoising sampling problems, yielding an exact symmetric formulation equivalent to the original sampler.The reformulation preserves modular prior-likelihood computation and the sampler's theoretical guarantees.Source: §1 · p. 1
• ODE diffusion implements the denoising updates, with the pretrained prior denoiser reused and the likelihood denoiser as the only problem-specific learned component.Source: §1 · pp. 1–2
• The likelihood denoiser uses known likelihood structure to derive a lightweight deep-unfolded network from a model-based iterative solver.Source: §1 · p. 2
• On nonlinear phase retrieval, learned likelihood updates achieve reconstruction quality comparable to MCMC-based updates with a 4.98× speedup in likelihood-update computation.Source: §1 · p. 2

2.1. Standard Split Gibbs Posterior Sampling

• The P-step and L-step alternately sample the augmented split-Gibbs conditionals, separating prior and likelihood computations through the auxiliary signal S′.A pretrained diffusion model can implement the prior update, while conventional MCMC can handle the likelihood update.Source: §2.1 · p. 2
• Under suitable regularity conditions, exact conditional updates and a sufficiently slow decrease of τ_i make the distribution of S_i converge to the target posterior.This guarantee applies when both updates sample their respective conditionals exactly.Source: §2.1 · p. 2

2.2. Reformulating P- and L-Steps as Denoising Samplers

• Both P- and L-steps can be rewritten as Gaussian denoising sampling steps, producing a symmetric formulation that preserves split Gibbs’s modular structure and theoretical guarantees.The P-step is exactly a denoising sampling step; the L-step uses a surrogate posterior built with a fictitious Gaussian prior because the likelihood alone is not a probability density.Source: §2.2 · pp. 2–3
• The alternating-denoising iteration applies the two reformulated updates in sequence, using the same Gaussian-denoising construction for both conditionals.The likelihood update uses the surrogate posterior as the distribution underlying its denoising step.Source: §2.2 · pp. 2–3

3.1. ODE-Based Diffusion for Denoising Sampling

• The ODE-based formulation extends diffusion sampling from the prior update to the likelihood update, enabling a unified implementation of both Gibbs steps.The reverse-time process starts from a high-noise initialization and yields an approximate sample when simulated backward.Source: §3.1 · p. 3
• A pretrained diffusion denoiser implements the prior step, whereas the likelihood step requires a conditional denoiser that depends on the observation.The framework therefore reduces the two sampling steps to evaluations of separate prior and likelihood denoisers.Source: §3.1 · p. 3

3.2. Likelihood Denoiser Design via Deep Unfolding

• The likelihood denoiser uses a model-informed unfolded proximal-gradient architecture motivated by MAP estimation, rather than a generic black-box network.The MAP formulation motivates the architecture, while MSE training targets the conditional-mean denoiser.Source: §3.2 · pp. 3–4
• Unfolding K iterations into network layers makes the stepsizes and measurement-model-induced parameters learnable, while coordinate-wise separability permits a small MLP per coordinate.Synthetic samples from the surrogate model train the denoiser to predict S′ over the considered noise levels.Source: §3.2 · p. 4

4. EXPERIMENTS

• The comparison uses phase retrieval on FFHQ 256 × 256 images, with 22 outer iterations after warm starting for each implementation.The learned denoiser has six unfolded proximal-gradient layers and a four-layer MLP for the proximal step.Source: §4 · p. 4
• 4.98× speedup: 10 NFEs achieve comparable PSNR and SSIM to 1000 MCMC iterations for the learned likelihood update.At 22 outer iterations, 15 NFEs yield slightly higher PSNR and SSIM than 1400 MCMC iterations.Source: §4 · p. 4

5. CONCLUSION

• The framework replaces likelihood-side MCMC with a lightweight model-informed deep-unfolded denoiser while implementing both Gibbs updates through ODE diffusion.Experiments on nonlinear phase retrieval demonstrate its effectiveness as an alternative to MCMC-based split Gibbs and its potential to reduce likelihood-update cost.Source: §5 · p. 4