Fractional Diffusion Bridge Models

2025-11-03

AI Summary

Standard diffusion and bridging models, grounded in Brownian motion, fail to capture memory effects, long-range dependencies, path roughness, and anomalous diffusion inherent in real stochastic processes. To address this, we propose the Fractional Diffusion Bridge Model (FDBM), the first diffusion bridge framework incorporating a non-Markovian, computationally tractable approximation of fractional Brownian motion (MA-fBM). We rigorously prove the existence of coupling-preserving generative bridges under this model and extend it to the Schrödinger bridge problem, deriving a principled loss function for unpaired data translation. Our method integrates fractional calculus with an efficient Markovian approximation scheme, enabling scalable training and inference. Experiments demonstrate significant improvements: reduced C$_alpha$ root-mean-square deviation in protein conformation prediction and lower Fréchet Inception Distance versus Brownian baselines in unpaired image translation—validating FDBM’s ability to model temporal correlations and anomalous diffusion.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Web Mining and Content Analysis: Content-based information diffusion

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.

Problem

• Modeling memory effects and long-range dependencies in stochastic processes
• Addressing anomalous diffusion phenomena beyond Brownian motion limitations
• Enabling tractable inference while preserving non-Markovian nature of fBM

Innovation

• Fractional Brownian motion approximates non-Markovian diffusion bridge
• Markovian approximation enables tractable inference for FDBM
• FDBM solves Schrödinger bridge problem for unpaired translation