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Fractional Diffusion Bridge Models

Conference: NeurIPS 2025 arXiv: 2511.01795 Code: GitHub-paired / GitHub-unpaired / SBFlow Area: Medical Imaging Keywords: Diffusion bridge models, fractional Brownian motion, protein conformation prediction, image translation, Schrödinger bridge

TL;DR

This paper proposes Fractional Diffusion Bridge Models (FDBM), which incorporate fractional Brownian motion (fBM) into the generative diffusion bridge framework. The Hurst exponent \(H\) controls the roughness and long-range dependence of trajectories, yielding improvements over Brownian motion baselines on protein conformation prediction and image translation tasks.

Background & Motivation

Background: Diffusion bridge models construct stochastic interpolation paths between two distributions by conditioning stochastic processes, and are widely applied to paired/unpaired data translation tasks such as protein conformation prediction and image translation. Existing models uniformly employ standard Brownian motion (BM) as the driving noise.

Limitations of Prior Work: Standard BM is a Markov process with independent increments, and thus cannot capture memory effects, long-range dependence, roughness, or anomalous diffusion present in real data. For complex systems such as proteins, the assumption of temporally independent increments may lead to insufficient modeling.

Key Challenge: The motivation for choosing BM in existing work is mathematical convenience rather than physical fidelity — adopting a more expressive driving noise should better match the true data dynamics.

Goal: To introduce fractional Brownian motion (non-Markovian, with long-range correlations) into diffusion bridges while maintaining trainability and efficient inference.

Key Insight: The paper exploits the Markovian approximation of fBM (MA-fBM), which approximates fBM via a linear superposition of \(K\) Ornstein-Uhlenbeck (OU) processes, making the augmented system Markovian.

Core Idea: Replace BM-driven diffusion bridges with the Markovian approximation of fBM, using the Hurst exponent \(H\) to flexibly control the properties of generated trajectories, improving performance in both paired and unpaired settings.

Method

Overall Architecture

The core of FDBM is to replace the BM in standard diffusion bridges with a scaled MA-fBM \(X = \sqrt{\varepsilon}\hat{B}^H\), where \(\hat{B}^H_t = \sum_{k=1}^K \omega_k Y_t^k\) is a weighted sum of \(K\) OU processes. The reference process \(X\) itself is non-Markovian, but the augmented process \(Z = (X, Y^1, \ldots, Y^K)\) is Markovian.

Key Designs

1. MA-fBM as Driving Noise

Type II fBM definition: \(B_t^H = \frac{1}{\Gamma(H+1/2)} \int_0^t (t-s)^{H-1/2} dB_s\)

  • \(H > 0.5\): positively correlated increments (super-diffusion), smoother trajectories
  • \(H < 0.5\): negatively correlated increments (sub-diffusion), rougher trajectories
  • \(H = 0.5\): reduces to standard BM

MA-fBM: \(K\) OU processes with rates \(\gamma_k = r^{k-n}\) are selected; optimal weights \(\omega\) are given by the closed-form linear system \(A\omega = b\), minimizing the \(L^2(\mathbb{P})\) error. Experiments fix \(K=5\).

2. MA-fBB (Markovian Approximation Fractional Brownian Bridge)

Conditioning the augmented process \(Z\) on its endpoints yields a partially pinned process \(Z_{|x_0,x_1}\), whose SDE drift includes a guidance term \(u(t,z)\) that is analytically computable. The conditional mean \(\mu_{1|t}(z)\) and conditional variance \(\sigma^2_{1|t}\) are both available in closed form.

3. Paired Data Translation

Proposition 5 (Coupling Preservation): There exists a stochastic process \(Z^\star\) that preserves the training data coupling \(\Pi_{0,1}\), satisfying \((X_0^\star, X_1^\star) \sim \Pi_{0,1}\).

Training objective: $\(\mathcal{L}_{\text{FDBM}}^{\text{paired}}(\theta) = \int_0^1 \mathbb{E}_{\mathbb{P}^\star}\left[\left\|\frac{X_1^\star - \mu_{1|t}(Z_t^\star)}{\sigma_{1|t}^2} - \tilde{u}^\theta(t, X_0, \mu_{1|t}(Z_t^\star))\right\|^2\right] dt\)$

Key point: The output dimension of the neural network \(\tilde{u}^\theta\) matches the data dimension \(d\) (not the augmented dimension), and is mapped to the augmented space via a scaling transformation. FDBM can therefore reuse the network architecture of ABM with negligible additional computational overhead.

4. Unpaired Data Translation (Schrödinger Bridge)

The reference process in the Schrödinger bridge (SB) problem is replaced with MA-fBM, defining the SB problem over the augmented space, and introducing augmented reciprocal classes and augmented Markovian projections. An \(\alpha\)-IMF training scheme (pre-training + fine-tuning) is adopted.

Loss & Training

  • Paired: Minimize the \(L^2\) distance between neural network predictions and the fractional bridge drift targets.
  • Unpaired: An analogous objective that does not condition on the starting value \(X_0\), using a forward-forward training strategy.
  • Note: In the unpaired setting, the MA-fBM reference process becomes unstable during fine-tuning when \(H\) deviates substantially from 0.5.

Key Experimental Results

Paired Data: Protein Conformation Prediction (D3PM Dataset)

Method Median RMSD(Å)↓ Mean RMSD(Å)↓ RMSD<2Å(%)↑ RMSD<5Å(%)↑ Δ RMSD Mean↑
SBALIGN 3.67 4.82 0% 71% 1.92
Sesame 2.87 3.65 38% 82% 3.11
ABM (baseline) 2.40 3.49 43% 84% 3.35
FDBM (H=0.2) 2.12 3.34 48% 86% 3.39
FDBM (H=0.3) 2.33 3.42 43% 85% 3.49
FDBM (H=0.1) 2.20 3.44 46% 83% 3.45

Unpaired Data: AFHQ Image Translation

Evaluation of cat↔wild translation on AFHQ-256 and AFHQ-512:

Method/Setting FID (wild→cat)↓ FID (cat→wild)↓
SBFlow (AFHQ-256) baseline baseline
FDBM (H=0.6, AFHQ-256) 19.42 11.62
FDBM (H=0.4, AFHQ-512) 30.11 14.27

Synthetic Data: Moons & T-Shape

Dataset ABM WSD FDBM Best WSD
Moons 0.015±0.019 0.012±0.002 (H=0.7)
T-Shape 0.082±0.028 0.048±0.039 (H=0.2)

Ablation Study

  • Choice of \(H\) is task-dependent: Moons benefits from smooth trajectories (\(H=0.6\)–\(0.7\)), while T-Shape and protein tasks favor rough trajectories (\(H=0.1\)–\(0.3\)).
  • \(K=5\) OU processes are sufficient; increasing \(K\) yields marginal improvements.
  • Diffusion coefficient \(\sqrt{\varepsilon}\): the optimal value for the protein task is 0.2.

Key Findings

  1. Rough trajectories (\(H<0.5\)) are superior for protein prediction: the proportion of RMSD<2Å increases from 43% (ABM) to 48%, and median RMSD decreases from 2.40Å to 2.12Å.
  2. ABM is already a strong baseline, outperforming all prior methods (SBALIGN, Sesame).
  3. FDBM preserves coupling (Proposition 5), unlike SBALIGN.
  4. Fine-tuning instability in the unpaired setting when \(H\) deviates far from 0.5, indicating forward-backward asymmetry challenges in the fBM SB problem.
  5. Minimal computational overhead: FDBM adds only the MA-fBM input/output transformations relative to ABM.

Highlights & Insights

  • This is the first work to introduce fBM into diffusion bridge modeling — conceptually elegant, as BM is merely the special case \(H=0.5\), and \(H\) can now be tuned to match data characteristics.
  • The theoretical proof of coupling preservation (Proposition 5) is a significant theoretical contribution.
  • Experimental findings suggest that protein conformational changes suit rough trajectories, implying memory effects in molecular motion.
  • The architecture reuse design is elegant: the neural network dimensionality is unchanged, with only a scaling mapping applied, minimizing implementation complexity.
  • The finding that Moons prefers smooth \(H\) while T-Shape prefers rough \(H\) suggests that the optimal \(H\) has an interpretable structural basis.

Limitations & Future Work

  1. Instability when \(H\) deviates far from 0.5 in the unpaired setting: SB fine-tuning fails to converge, limiting the flexibility of FDBM in unpaired tasks.
  2. \(H\) requires manual search: no automatic method for selecting \(H\) currently exists; treating it as a learnable parameter is a natural direction.
  3. MA-fBM is an approximation: the \(K=5\) approximation has limited precision and may be insufficient in certain scenarios.
  4. Lack of adaptive \(H\): different regions may require different \(H\) values, and a globally fixed \(H\) may be suboptimal.
  5. Protein experiments are conducted only on a subset of D3PM: larger-scale validation is needed.
  • Builds upon ABM (Bortoli et al.), SBFlow (Bortoli et al.), and SBALIGN (Somnath et al.).
  • The MA-fBM technique derives from Harms & Stefanovits and Daems et al.
  • Core insight: the statistical properties of the driving noise (memory, roughness) should be matched to the physical characteristics of the target data, rather than chosen solely for mathematical convenience.

Rating

⭐⭐⭐⭐ (4/5)

Rationale: The work is conceptually novel (the first to introduce fBM into diffusion bridges), theoretically sound (coupling preservation proof), and experimentally comprehensive (synthetic + protein + image). The improvements on protein tasks are meaningful. The instability in the unpaired setting is a clear limitation.