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Generative Fractional Diffusion Models
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Introduces generative fractional diffusion models (GFDM), which drive score-based diffusion with a Markov-approximate fractional Brownian motion and learn the score with a data-dimensional network via augmented score matching.
Problem
Standard continuous-time score-based diffusion models use Brownian motion, a light-tailed process with independent increments, which is linked to slow convergence, mode collapse on imbalanced data, and limited diversity; fractional Brownian motion offers correlated increments but lacks a tractable r
Approach
Replace the driving Brownian motion with a Markov approximation of type II fBM (a weighted sum of Ornstein-Uhlenbeck processes), derive a continuous reparameterization for the conditional forward marginals, and use augmented score matching so that only a data-dimensional score network is learned while the known OU score is used as guidance.
Key results
- On MNIST, FVP with H=0.9 and K=3 reaches FID 0.72 versus 1.44 for retrained BM-driven VP.
- On CIFAR10, FVP(H=0.9,K=2) achieves FID 3.77 versus 4.85 for retrained VP, with pixel Vendi score 3.60 versus 3.28.
- Super-diffusive regime H=0.9 yields higher class-wise Recall than VP across all CIFAR10 classes, though the PF ODE with BM dynamics has better FID.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace Brownian VP dynamics with FVP dynamics (H=0.9, K=3) | MNIST image generation · MNIST | FID: 1.44 -> 0.72 | Helped |
| replace Brownian VP dynamics with FVP dynamics (H=0.9, K=3) | MNIST image generation · MNIST | pixel Vendi Score: 23.64 to 24.18 | Helped |
| replace Brownian VP dynamics with FVP dynamics (H=0.9, K=2) | CIFAR10 image generation · CIFAR10 | FID: 4.85 -> 3.77 | Helped |
| replace Brownian VP dynamics with FVP dynamics (H=0.9, K=2) | CIFAR10 image generation · CIFAR10 | pixel Vendi Score: 3.28 to 3.60 | Helped |
| replace Brownian VP dynamics with FVP dynamics (H=0.9, K=2) | CIFAR10 image generation · CIFAR10 | Recall: 0.6814 -> 0.7038 | Helped |
| increase augmenting processes from K=1 to K=5 in FVE dynamics at H=0.5 (ablation) | MNIST image generation · MNIST | pixel Vendi Score: 24.20 to 23.08 | Helped |
| increase augmenting processes from K=1 to K=5 in FVP dynamics at H=0.5 (ablation) | MNIST image generation · MNIST | pixel Vendi Score: 23.69 to 24.56 | Helped |
| increase augmenting processes from K=1 to K=5 in FVP dynamics at H=0.5 (ablation) | MNIST image generation · MNIST | NLLs Test: 3.90 -> 7.09 | Hurt |
| increase K from 2 to 5 at H=0.9 (ablation) | CIFAR10 FVP · CIFAR10 | FID: 3.77 -> 69.06 | Hurt |
| use H=0.5 instead of H=0.9 for FVP with K=2 (ablation) | CIFAR10 image generation · CIFAR10 | FID: 3.77 -> 4.85 | Hurt |
| vary K among augmenting-process counts (ablation) | FVP image generation · MNIST and CIFAR10 | FID | Mixed |
Surprising findings
- Larger numbers of augmenting processes (K=5) degrade FID sharply on CIFAR10 for some Hurst indices, e.g. FID 96.54 at H=0.5 with K=5.
- For the PF ODE sampler, the purely Brownian VP dynamics outperform the super-diffusive regime in FID.
Limitations
- Evaluation limited to MNIST and CIFAR10, so generalization to other datasets and modalities is not established.
- Type II fBM is used for experiments, the empirical study of type I fBM is left to future work.
- Performance is sensitive to K and H; large K can severely degrade FID, and the optimal H is not known.
Experimental setup
Datasets: MNIST CIFAR10
Benchmarks: MNIST CIFAR10
Metrics: FID Inception score pixel Vendi score improved recall NLL
Baselines: VE (retrained) VP (retrained) DDPM-style VP SDE PF ODE
Builds on: score-based SDE Markov approximation of fBM (Daems et al.) variance preserving SDE variance exploding SDE probability flow ODE
Closest papers by meaning
- Fractional Diffusion Bridge Models (NeurIPS 2025 · similarity 0.90)
- Towards Non-Asymptotic Convergence for Diffusion-Based Generative Models (ICLR 2024 · similarity 0.88)
- Variational Inference for SDEs Driven by Fractional Noise (ICLR 2024 · similarity 0.88)
- Distributional Diffusion Models with Scoring Rules (ICML 2025 · similarity 0.88)
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