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Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
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Introduces Feynman-Kac Correctors (FKCs), weighted SDEs plus SMC resampling that let pretrained diffusion models be sampled from annealed, geometric-averaged, or product distributions, applied to CFG, molecule generation, and Boltzmann samplers.
Problem
Controlling inference-time behavior of score-based diffusion models so that samples follow prescribed intermediate distributions (annealed, geometric-average, product of experts) rather than heuristic score mixing such as classifier-free guidance, which does not approximate intermediate marginals.
Approach
Derive weighted SDEs from Feynman-Kac PDEs by converting flow and diffusion terms into reweighting terms, yielding importance weights that make simulated particles properly weighted for annealed, geometric-averaged, or product targets; then use SMC systematic resampling along the trajectory.
Key results
- On EDM2-XS CFG image generation with 10,000 images, FKC (K=8 particles) improves ImageReward and CLIP score over vanilla CFG, e.g. CLIP 29.14 vs 28.75 and IR 0.05 vs -0.24 at gamma=40, N=32.
- For dual-target molecule generation (target score SDE, beta=2.0), FKC improves average product docking score from 71.186 to 77.271 (higher is better) and lowers max docking score from -7.421 to -7.720 vs no resampling.
- On LJ-13, tempered noise+FKC performs best at higher temperatures while target score SDE+FKC performs best at lower temperatures, both outperforming DEM trained directly at the lower temperature for T_S in [2.0, 0.8].
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add FKC resampling to EDM2+CFG (ablation) | EDM2-XS image generation · ImageNet-512 image generation (γ=40, N=32) | CLIP Score: 28.75 -> 29.00 | Helped |
| Add FKC resampling to EDM2+CFG (ablation) | EDM2-XS image generation · ImageNet-512 image generation (γ=40, N=32) | ImageReward: -0.24 -> 0.04 | Helped |
| Add FKC resampling to Target Score SDE (ablation) | Dual-target molecule generation at β=2.0 · Dual-target molecule generation | Maximum docking score: -7.421_{\pm 2.497} -> -7.720_{\pm 2.562} | Helped |
| Add FKC resampling to Target Score SDE (ablation) | Dual-target molecule generation at β=2.0 · Dual-target molecule generation | Better than reference: 0.383_{\pm 0.389} -> 0.450_{\pm 0.438} | Helped |
| Add FKC resampling to Target Score SDE (ablation) | Dual-target molecule generation at β=2.0 · Dual-target molecule generation | Diversity: 0.877_{\pm 0.015} -> 0.806_{\pm 0.048} | Hurt |
| Add FKC resampling to Target Score SDE (ablation) | Dual-target molecule generation at β=2.0 · Dual-target molecule generation | Validity and uniqueness: 0.961_{\pm 0.115} -> 0.862_{\pm 0.174} | Hurt |
| Add FKC resampling to Target Score SDE (ablation) | Dual-target molecule generation at β=1.0 · Dual-target molecule generation | Maximum docking score: -7.256_{\pm 2.622} -> -7.649_{\pm 2.605} | Helped |
| Add FKC resampling to Target Score SDE (ablation) | Dual-target molecule generation at β=0.5 · Dual-target molecule generation | Maximum docking score: -7.030_{\pm 2.556} -> -6.966_{\pm 3.291} | Hurt |
| Increase FKC particle count from K=2 to K=8 (ablation) | EDM2 image generation · ImageNet-512 image generation | Image quality | Helped |
| Use target-score SDE + FKC rather than tempered-noise SDE + FKC (ablation) | Annealed sampling from a mixture of 40 Gaussians · Mixture of 40 Gaussians | Sampling performance | Helped |
| Use target-score SDE + FKC rather than tempered-noise SDE + FKC at lower temperatures (ablation) | LJ-13 sampler trained at T_L=2 · LJ-13 temperature annealing | 2-Wasserstein | Helped |
| Use tempered-noise SDE + FKC rather than target-score SDE + FKC at higher temperatures (ablation) | LJ-13 sampler trained at T_L=2 · LJ-13 temperature annealing | 2-Wasserstein | Helped |
Surprising findings
- Final-step images from a single FKC batch are nearly identical visually due to weight degeneracy, even though metrics improve.
Limitations
- Resampling with K particles adds computational cost proportional to K.
- Weight degeneracy causes near-identical final samples in a batch.
- Diversity and uniqueness decrease when resampling with FKC in molecule generation.
Experimental setup
Datasets: ImageNet-512 CrossDocked
Benchmarks: CLIP Score ImageReward AutoDock Vina docking scores (100 protein pairs) Mixture of 40 Gaussians LJ-13 energy histograms
Metrics: CLIP score ImageReward docking score 2-Wasserstein distance diversity validity and uniqueness QED SA score
Baselines: classifier-free guidance TargetDiff DualDiff DEM trained at lower temperature
Builds on: classifier-free guidance denoising energy matching predictor-corrector sampling EDM2 TargetDiff
Closest papers by meaning
- Progressive Inference-Time Annealing of Diffusion Models for Sampling from Boltzmann Densities (NeurIPS 2025 · similarity 0.88)
- A General Framework for Inference-time Scaling and Steering of Diffusion Models (ICML 2025 · similarity 0.88)
- Diffusion Path Samplers via Sequential Monte Carlo (NeurIPS 2026 · similarity 0.88)
- Test-time Alignment of Diffusion Models without Reward Over-optimization (ICLR 2025 · similarity 0.87)
- Informed Correctors for Discrete Diffusion Models (NeurIPS 2025 · similarity 0.87)
- Reverse Diffusion Sequential Monte Carlo Samplers (NeurIPS 2025 · similarity 0.87)
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