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Workshop: NeurIPS 2023 Workshop on Diffusion Models

Generative Fractional Diffusion Models

Gabriel Nobis · Marco Aversa · Maximilian Springenberg · Michael Detzel · Stefano Ermon · Shinichi Nakajima · Roderick Murray-Smith · Sebastian Lapuschkin · Christoph Knochenhauer · Luis Oala · Wojciech Samek

Abstract: We generalize the continuous time framework for score-based generative models from an underlying Brownian motion (BM) to an approximation of fractional Brownian motion (FBM). We derive a continuous reparameterization trick and the reverse time model by representing FBM as a stochastic integral over a family of Ornstein-Uhlenbeck processes to define generative fractional diffusion models (GFDM) with driving noise converging to a non-Markovian process of infinite quadratic variation. The Hurst index $H\in(0,1)$ of FBM enables control of the roughness of the distribution transforming path. To the best of our knowledge, this is the first attempt to build a generative model upon a stochastic process with infinite quadratic variation.

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