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Discrete Spatial Diffusion: Intensity-Preserving Diffusion Modeling
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Proposes Discrete Spatial Diffusion, a continuous-time discrete-state random-walk diffusion model that exactly conserves per-channel particle counts in forward and reverse processes, applied to images and battery/rock microstructures.
Problem
Standard diffusion models operate on continuous intensities with independent per-pixel noise, so they cannot exactly preserve discrete, conserved quantities (e.g., particle counts, mass/phase fractions) needed in scientific generative modeling.
Approach
Replace independent per-pixel corruption with a spatially correlated jump process in which intensity units perform random walks on a lattice (nearest-neighbor hops), so total intensity per color channel is conserved exactly; the reverse rates are learned by a NCSN++ network with softplus outputs, and sampling uses binomial tau-leaping with a CFL-based adaptive step.
Key results
- On CIFAR-10, DSD with CFL 0.01 and bilateral filtering reaches FID 28.2 (sFID 16.4) on 50,000 samples, and 20.6 FID with classifier thresholding on 21,780 samples.
- On CelebA 64x64, DSD with filtering reaches FID 28.9 (sFID 25.7).
- Unlike conditional Gaussian diffusion, DSD exactly conserves total intensity per sample, while Gaussian baselines produce negative values and miss target intensity especially in distribution tails.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add bilateral filtering (ablation) | DSD on CIFAR-10 with CFL ε = 0.01 · CIFAR-10 | FID: 46.3 -> 28.2 | Helped |
| add bilateral filtering (ablation) | DSD on CIFAR-10 with CFL ε = 0.01 · CIFAR-10 | sFID: 23.4 -> 16.4 | Helped |
| add classifier thresholding (p ≥ 0.99) after bilateral filtering (ablation) | DSD on CIFAR-10 · CIFAR-10 | FID: 28.2 -> 20.6 | Helped |
| add classifier thresholding (p ≥ 0.99) after bilateral filtering (ablation) | DSD on CIFAR-10 · CIFAR-10 | sFID: 16.4 -> 17.7 | Hurt |
| add bilateral filtering (ablation) | DSD on CelebA 64×64 with CFL ε = 0.05 · CelebA 64×64 | FID: 44.2 -> 28.9 | Helped |
| add bilateral filtering (ablation) | DSD on CelebA 64×64 with CFL ε = 0.05 · CelebA 64×64 | sFID: 29.0 -> 25.7 | Helped |
| lower the CFL tolerance (ablation) | DSD sampling on CIFAR-10 · CIFAR-10 | FID | Helped |
| replace rate-matching loss with likelihood loss (ablation) | DSD training · image synthesis and scientific datasets | model performance | No clear effect |
Surprising findings
- Gaussian diffusion conditioned on total intensity produces small negative values and fails to respect conditioning in distribution tails, while DSD holds exactly by construction.
- Discrete pixel-hopping sampling introduces high-frequency fluctuations that degrade FID; a fixed bilateral filter substantially improves FID and sFID.
Limitations
- Computational cost of forward and reverse sampling scales linearly with total image intensity, making it less efficient for uint8 or high-intensity images.
- FID on CIFAR-10 and CelebA is not competitive with state-of-the-art models; generated samples show pixel-level high-frequency artifacts.
- Requires custom forward process code and a novel sampling scheme, a steeper learning curve than standard diffusion.
Experimental setup
Datasets: MNIST CIFAR-10 CelebA 64x64 Berea Sandstone X-ray microCT Savonnieres carbonate Massangis limestone lithium-ion NMC electrode tomography
Benchmarks: CIFAR-10 CelebA 64x64 MNIST
Metrics: FID sFID SSIM intensity error two-point correlation pore size distribution
Baselines: Gaussian diffusion with conditioning discretized Gaussian diffusion guidance function approach (MSE penalty)
Builds on: NCSN++ Inverse Heat Dissipation Model Blurring Diffusion Model binomial tau-leaping Cold Diffusion
Closest papers by meaning
- One-step Diffusion with Distribution Matching Distillation (CVPR 2024 · similarity 0.87)
- Fast Solvers for Discrete Diffusion Models: Theory and Applications of High-Order Algorithms (NeurIPS 2025 · similarity 0.87)
- Why Masking Diffusion Works: Condition on the Jump Schedule for Improved Discrete Diffusion (NeurIPS 2025 · similarity 0.87)
- Distillation of Discrete Diffusion through Dimensional Correlations (ICML 2025 · similarity 0.87)
- Split Gibbs Discrete Diffusion Posterior Sampling (NeurIPS 2025 · similarity 0.87)
- Diffusion Model is Effectively Its Own Teacher (CVPR 2025 · similarity 0.87)
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