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Solving Linear-Gaussian Bayesian Inverse Problems with Decoupled Diffusion Sequential Monte Carlo
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Proposes DDSMC, an asymptotically exact sequential Monte Carlo method for linear-Gaussian Bayesian inverse problems with diffusion priors, built on decoupled diffusion (DAPS), extended to discrete data as D3SMC.
Problem
Posterior sampling for linear-Gaussian Bayesian inverse problems using pre-trained diffusion models as priors, where existing approximate methods lack consistency guarantees and struggle on simple tasks.
Approach
Uses the DAPS decoupled-diffusion transition (reconstruct x0 then re-noise with the forward kernel) as the SMC prior, generalized by an inverse temperature eta, with a Gaussian-approximated posterior-based proposal and likelihood targets so that SMC targets the exact posterior induced by the chosen prior.
Key results
- On the GMM experiment with d_x=800, d_y=4, DDSMC achieves sliced Wasserstein 0.16±0.10 versus 1.21±0.94 for MCGDiff and 9.21±0.83 for DAPS.
- On FFHQ, DDSMC-1.0 (Tweedie) gets LPIPS 0.14 on SR4 versus 0.10 for DCPS and 0.15 for DAPS and MCGDiff; DDSMC is on par with prior methods rather than best.
- DDSMC with a single particle is essentially equivalent to DAPS, so gains come from SMC resampling with multiple particles.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use DDSMC instead of MCGDiff | posterior sampling · GMM, all reported dimensions | sliced Wasserstein distance: 1.79±0.54 -> 0.88±0.33; 0.80±0.41 -> 0.34±0.20; 0.26±0.21 -> 0.09±0.08; 1.06±0.36 -> 0.50±0.16; 1.04±0.49 -> 0.37±0.20; | Helped |
| use DDSMC instead of DCPS | FFHQ image restoration · middle inpainting, half outpainting, and 4× super-resolution | LPIPS: 0.03 -> 0.05; 0.20 -> 0.23; 0.10 -> 0.14 | Hurt |
| replace Tweedie's reconstruction with PF-ODE reconstruction (ablation) | DDSMC · GMM (d_x=800, d_y=1), η=0.0 | sliced Wasserstein distance: 1.73±0.54 -> 1.37±0.43 | Helped |
| increase η from 0.0 to 0.5 (ablation) | DDSMC with Tweedie's reconstruction · GMM (d_x=800, d_y=1) | sliced Wasserstein distance: 1.73±0.54 -> 1.64±0.61 | Helped |
| increase η from 0.5 to 1.0 (ablation) | DDSMC with Tweedie's reconstruction · GMM (d_x=800, d_y=1) | sliced Wasserstein distance: 1.64±0.61 -> 1.86±0.63 | Hurt |
| increase η from 0.0 to 0.5 (ablation) | DDSMC with PF-ODE reconstruction · GMM (d_x=800, d_y=1) | sliced Wasserstein distance: 1.37±0.43 -> 1.68±0.52 | Hurt |
| increase η from 0.5 to 1.0 (ablation) | DDSMC with PF-ODE reconstruction · GMM (d_x=800, d_y=1) | sliced Wasserstein distance: 1.68±0.52 -> 2.09±0.51 | Hurt |
| increase η from 0.0 to 0.5 (ablation) | DDSMC with Tweedie's reconstruction · FFHQ super-resolution 4× | LPIPS: 0.27 -> 0.20 | Helped |
| increase η from 0.5 to 1.0 (ablation) | DDSMC with Tweedie's reconstruction · FFHQ super-resolution 4× | LPIPS: 0.20 -> 0.14 | Helped |
| increase η from 0.5 to 1.0 (ablation) | DDSMC with PF-ODE reconstruction · FFHQ super-resolution 4× | LPIPS: 0.15 -> 0.36 | Hurt |
| increase the number of particles (ablation) | DDSMC · protein structure completion | RMSD | Helped |
Surprising findings
- On LPIPS image metrics, DCPS outperforms DDSMC, while DDSMC is far better on the synthetic GMM posterior task, showing perceptual quality and posterior accuracy disagree.
Limitations
- Image and protein results are only on par with prior methods; LPIPS does not measure posterior approximation quality.
- Gaussian approximation of the posterior p(x0|xt) introduces approximations in the proposal and intermediate targets.
- TDS showed instability and crashes in the GMM experiments, so fewer runs were used for it.
Experimental setup
Datasets: Gaussian mixture model (synthetic) FFHQ ATAD2 protein (PDB 7qum) binarized MNIST
Benchmarks: Gaussian mixture posterior (sliced Wasserstein) FFHQ inpainting/outpainting/SR4 (LPIPS) protein structure completion (RMSD) binarized MNIST denoising
Metrics: sliced Wasserstein distance LPIPS PSNR RMSD
Baselines: MCGDiff TDS DDRM DCPS DAPS ADP-3D
Builds on: DAPS MCGDiff TDS DDIM D3PM
Closest papers by meaning
- Inverse Problem Sampling in Latent Space Using Sequential Monte Carlo (ICML 2025 · similarity 0.90)
- Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors (NeurIPS 2024 · similarity 0.89)
- Preconditioned Langevin Dynamics with Score-based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problems (NeurIPS 2025 · similarity 0.89)
- Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing (CVPR 2025 · similarity 0.88)
- Divide-and-Conquer Posterior Sampling for Denoising Diffusion priors (NeurIPS 2024 · similarity 0.88)
- Learning Diffusion Priors from Observations by Expectation Maximization (NeurIPS 2024 · similarity 0.88)
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