Fractional Diffusion Bridge Models

TL;DR

This work extends diffusion-bridge models by replacing standard Brownian noise with a Markov-approximate fractional Brownian motion (MA-fBM) to capture memory, roughness, and long-range dependencies observed in real data. By augmenting MA-fBM with Ornstein–Uhlenbeck processes, the authors construct a Markovian MA-fBB that yields a coupling-preserving stochastic bridge for paired data and a Schrödinger-bridge formulation for unpaired data, with learning driven by neural approximations of intractable drifts. The resulting Fractional Diffusion Bridge Models (FDBM) demonstrate improved performance on two fronts: predicting protein conformational changes with lower RMSD and performing unpaired image translation with better Fréchet Inception Distance (FID) and perceptual similarity across high-dimensional domains. The framework supports a principled training loss for both paired and unpaired settings and provides publicly available implementations, laying the groundwork for memory-aware generative diffusion in scientific and vision tasks.

Abstract

We present Fractional Diffusion Bridge Models (FDBM), a novel generative diffusion bridge framework driven by an approximation of the rich and non-Markovian fractional Brownian motion (fBM). Real stochastic processes exhibit a degree of memory effects (correlations in time), long-range dependencies, roughness and anomalous diffusion phenomena that are not captured in standard diffusion or bridge modeling due to the use of Brownian motion (BM). As a remedy, leveraging a recent Markovian approximation of fBM (MA-fBM), we construct FDBM that enable tractable inference while preserving the non-Markovian nature of fBM. We prove the existence of a coupling-preserving generative diffusion bridge and leverage it for future state prediction from paired training data. We then extend our formulation to the Schrödinger bridge problem and derive a principled loss function to learn the unpaired data translation. We evaluate FDBM on both tasks: predicting future protein conformations from aligned data, and unpaired image translation. In both settings, FDBM achieves superior performance compared to the Brownian baselines, yielding lower root mean squared deviation (RMSD) of C$_α$ atomic positions in protein structure prediction and lower Fréchet Inception Distance (FID) in unpaired image translation.

Figures (12)

• Figure 1: Trajectories from the approximate $2d$-fractional Brownian bridge for different Hurst indices $H$.
• Figure 2:
• Figure 3: Qualitative comparison on Moons and T-Shape. Plots and datasets design follow somnath2023aligned.
• Figure 4: D3PM Conformational changes, results marked with an asterisk ($^{\star}$) are obtained from the specified reference. Metrics for FDBM and ABM are averaged over $5$ training trials.
• Figure 5: Exemplary FDBM samples (ours) for wild $\rightarrow$ cat (a, b) and cat $\rightarrow$ wild (c, d) using DiT-L/2 on AFHQ-512 and AFHQ-256. Left: inputs; right: Euler--Maruyama samples (distinct seeds).

Theorems & Definitions (19)

• Definition 1: Type ii Fractional Brownian motion levy1953random
• Definition 2: Markov approximation of fBM HARMS2019daems2023variational
• Proposition 3: Optimal Approximation Coefficients daems2023variational
• Proposition 4: Markov approximation of a fractional Brownian bridge daems2025phdthesisdaems2025efficient
• Proposition 5
• Definition 6
• Definition 7
• Proposition 8
• proof
• Proposition 9: Markov approximation of a fractional Brownian bridge daems2025efficient