Fractional Diffusion Bridge Models

TL;DR

The paper introduces Fractional Diffusion Bridge Models (FDBM), a new framework that captures non-Markovian memory effects in stochastic processes, improving predictions in protein structures and image translation over traditional Brownian models.

Contribution

It develops a Markovian approximation of fractional Brownian motion to create tractable non-Markovian diffusion bridges, extending to Schrödinger bridges for unpaired data translation.

Findings

FDBM outperforms Brownian baselines in protein conformation prediction.

FDBM achieves lower FID scores in unpaired image translation.

The model effectively captures long-range dependencies in data.

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 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 Fr\'echet Inception Distance (FID) in unpaired image translation.