Fractional Diffusion Bridge Models
Gabriel Nobis, Maximilian Springenberg, Arina Belova, Rembert Daems, Christoph Knochenhauer, Manfred Opper, Tolga Birdal, Wojciech Samek
We present *Fractional Diffusion Bridge Models* (FDBM), a novel generative diffusion bridge framework driven by 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. As a remedy, leveraging a recent Markovian approximation (MA-fBM), we construct FDBMs that enable tractable inference while preserving the non-Markovian nature of fBM. We prove the resulting bridge is a coupling-preserving process and leverage it for future state prediction from paired training data. We then extend our formulation to the Schr\"{o}dinger 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$_\alpha$ atomic positions in protein structure prediction and lower Fréchet Inception Distance (FID) in unpaired image translation.