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
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.