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.

翻译：我们将基于分数布朗运动（FBM）逼近的连续时间框架，从基础布朗运动（BM）推广至基于分数的生成模型。通过将FBM表示为奥恩斯坦-乌伦贝克过程族的随机积分，推导出连续重参数化技巧与逆向时间模型，从而定义生成式分数阶扩散模型（GFDM），其驱动噪声收敛至具有无穷二次变分的非马尔可夫过程。FBM的赫斯特指数 $H\in(0,1)$ 可控制分布变换路径的粗糙度。据我们所知，这是首个基于无穷二次变分随机过程构建生成模型的研究。