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Predict-Project-Renoise: Sampling Diffusion Models under Hard Constraints
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Proposes Predict-Project-Renoise (PPR), a corrector-kernel sampler that projects denoiser outputs onto hard constraints and renoises iteratively to sample pretrained diffusion models from the constrained marginal.
Problem
Pretrained diffusion models cannot enforce hard constraints (conservation laws, boundary conditions, observation consistency) at sampling time, and existing projection or posterior-sampling methods either violate constraints, leave the data manifold, or sample the wrong constrained distribution.
Approach
Identify a corrector kernel whose unique stationary distribution is the constrained marginal at each noise level; approximate it by projecting the denoiser output (minimizing c(d_theta(x_t)) w.r.t. x_t), taking the denoised point estimate, and renoising with the forward kernel, repeated M times per reverse step.
Key results
- On KS, PPR reaches 82.7% feasibility vs 1% for the next best method, with CRPS 0.177 vs 0.206 for TS.
- On Data2D, PPR reaches 79.0% feasibility vs 31.6% for CPS, with k-NN cross-edge rate 0.375 (near 0.5 ideal).
- On Appa weather, PPR achieves the lowest skill, CRPS and spread for 2m temperature (skill 1.399, CRPS 0.693) versus MMPS (1.536, 0.708).
Surprising findings
- Without renoising (M=0) the KS sampler diverged, and projecting in x0-space without the denoiser yielded no feasible KS samples.
- Projecting the clean estimate directly (x0-space) produces discontinuous samples outside the data support.
Limitations
- Requires a differentiable constraint function; non-differentiable, discrete or black-box constraints are not handled.
- Approximates the constrained posterior by a point estimate, introducing irreducible bias.
- Projection through the denoiser with differentiation adds compute cost.
Experimental setup
Datasets: ERA5 Kuramoto-Sivashinsky simulated trajectories Data2D checkerboard and banana-GMM
Benchmarks: Data2D Kuramoto-Sivashinsky Appa weather forecasting (WeatherBench2)
Metrics: feasibility % constraint violation skill RMSE CRPS spread max-norm k-NN cross-edge rate wall time
Baselines: DPS MMPS TS (Trust Sampling) CPS PDM HDC (ReSample)
Builds on: predictor-corrector sampler DDPM EDM Tweedie's formula DPS
Closest papers by meaning
- DiffDA: a Diffusion model for weather-scale Data Assimilation (ICML 2024 · similarity 0.86)
- Constrained Diffusion Models via Dual Training (NeurIPS 2024 · similarity 0.86)
- Fast constrained sampling in pre-trained diffusion models (NeurIPS 2025 · similarity 0.86)
- Free Hunch: Denoiser Covariance Estimation for Diffusion Models Without Extra Costs (ICLR 2025 · similarity 0.86)
- Score-Optimal Diffusion Schedules (NeurIPS 2024 · similarity 0.85)
- Interpreting and Improving Diffusion Models from an Optimization Perspective (ICML 2024 · similarity 0.85)
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