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Fractional Diffusion Bridge Models
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Proposes Fractional Diffusion Bridge Models (FDBM), which build paired and unpaired diffusion/Schrödinger bridges on a Markov-approximate fractional Brownian motion reference, and reports gains over Brownian baselines on protein conformational change prediction and AFHQ cat-wild image translation.
Problem
Standard diffusion bridges use Brownian motion, which is Markovian with independent increments and cannot capture memory, long-range dependence, or roughness in real stochastic data such as protein conformational dynamics or image translation.
Approach
Replace the Brownian reference in augmented bridge matching and Schrödinger bridge flow with a scaled Markov-approximate fBM (a weighted sum of K OU processes driven by one BM), giving a tractable fractional bridge with closed-form drift terms controlled by the Hurst index H.
Key results
- On D3PM conformational changes, FDBM (H=0.2) reaches median RMSD 2.12 Å versus 2.40 Å for retrained ABM, and 48% versus 43% of predictions below 2 Å.
- On AFHQ-32 cats→wild with DiT-L/2, FDBM with H=0.5 reaches FID 20.26 after online finetuning versus 33.92 for SBFlow DiT-L/2.
- On AFHQ-256 wild/cats with DiT-L/2 in pretraining, FDBM (H=0.6) reaches FID 11.62 (cats→wild) and 19.42 (cats←wild), versus SBFlow DiT-L/2 at 16.62 and 33.96.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the Brownian bridge baseline ABM with FDBM at H=0.2 | Protein conformation prediction · D3PM Test Set | Median RMSD (Å): 2.40 -> 2.12 | Helped |
| Use FDBM with H=0.2 instead of ABM | Protein conformation prediction · D3PM Test Set | % RMSD (Å)<2: 43% -> 48% | Helped |
| Use FDBM with H=0.3 instead of ABM | Protein conformation prediction · D3PM Test Set | Mean ΔRMSD (Å): 3.35 -> 3.49 | Helped |
| Replace SBFlow with FDBM (DiT-L/2) | Unpaired image translation · AFHQ-32 cats→wild online finetuning | FID: $33.92$ -> $20.26$ | Helped |
| Replace SBFlow with FDBM (DiT-L/2) | Unpaired image translation · AFHQ-32 wild→cats online finetuning | FID: $54.10$ -> $26.79$ | Helped |
| Replace SBFlow with FDBM (DiT-L/2) | Unpaired image translation · AFHQ-256 cats→wild pretraining | FID: $16.62$ -> $11.62$ | Helped |
| Replace SBFlow with FDBM (DiT-L/2) | Unpaired image translation · AFHQ-256 wild→cats pretraining | FID: $33.96$ -> $19.42$ | Helped |
| Vary the Hurst index below 0.3 (ablation) | Unpaired FDBM · AFHQ image translation | generation stability | Hurt |
| Increase the number of OU processes above 5 (ablation) | Unpaired FDBM · AFHQ image translation | generation stability | Hurt |
| Use online finetuning with MA-fBM at H=0.5 (ablation) | Unpaired FDBM · AFHQ image translation | FID | Helped |
| Use online finetuning with MA-fBM at H other than 0.5 (ablation) | Unpaired FDBM · AFHQ image translation | FID | No clear effect |
Surprising findings
- Online finetuning improved FDBM only for H=0.5, not for other Hurst indices.
- The method is unstable for K>5 and H<0.3 in unpaired translation.
Limitations
- Unpaired finetuning stage performs reliably only near H=0.5; no convergence guarantee to the Schrödinger bridge is claimed.
- Gains on the D3PM benchmark are modest (e.g., one metric slightly worse than baseline) and based on five trials.
- Theoretical results rely on assumptions (absolute continuity, regularity of h), and the reciprocal-class conjecture is unproven.
Experimental setup
Datasets: D3PM AFHQ Moons T-Shape
Benchmarks: D3PM test set AFHQ-32 AFHQ-256 AFHQ-512 Moons T-Shape
Metrics: RMSD percent RMSD below threshold delta RMSD FID LPIPS Wasserstein-1 distance
Baselines: ABM SBALIGN Sesame EGNN SBFlow
Builds on: Augmented Bridge Matching Schrödinger Bridge Flow Markov approximation of fBM Doob h-transform DiT
Closest papers by meaning
- Generative Fractional Diffusion Models (NeurIPS 2024 · similarity 0.90)
- Diffusion Bridge Implicit Models (ICLR 2025 · similarity 0.89)
- Learning non-equilibrium diffusions with Schrödinger bridges: from exactly solvable to simulation-free (NeurIPS 2025 · similarity 0.88)
- Go With the Flow: Fast Diffusion for Gaussian Mixture Models (NeurIPS 2025 · similarity 0.88)
- Finding Local Diffusion Schrodinger Bridge using Kolmogorov-Arnold Network (CVPR 2025 · similarity 0.87)
- Discrete Diffusion Schrödinger Bridge Matching for Graph Transformation (ICLR 2025 · similarity 0.87)
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