Score-based diffusion models generate samples from an unknown target distribution using a time-reversed diffusion process. While such models represent state-of-the-art approaches in industrial applications such as artificial image generation, it has recently been noted that their performance can be further improved by considering injection noise with heavy tailed characteristics. Here, I present a generalization of generative diffusion processes to a wide class of non-Gaussian noise processes. I consider forward processes driven by standard Gaussian noise with super-imposed Poisson jumps representing a finite activity Levy process. The generative process is shown to be governed by a generalized score function that depends on the jump amplitude distribution and can be estimated by minimizing a simple MSE loss as in conventional Gaussian models. Both probability flow ODE and SDE formulations are derived using basic technical effort. A detailed implementation for a pure jump process with Laplace distributed amplitudes yields a generalized score function in closed analytical form and is shown to outperform the equivalent Gaussian model in specific parameter regimes.

翻译：基于分数的扩散模型通过时间反转的扩散过程从未知目标分布中生成样本。虽然此类模型在人工图像生成等工业应用中代表了最先进的方法，但最近有研究指出，通过考虑具有重尾特性的注入噪声，其性能可得到进一步提升。本文提出将生成扩散过程推广至一大类非高斯噪声过程。我考虑由标准高斯噪声叠加泊松跳跃（代表有限活动Lévy过程）驱动的正向过程。研究表明，生成过程受广义分数函数控制，该函数依赖于跳跃幅度分布，并可通过最小化简单均方误差损失进行估计，这与传统高斯模型类似。通过基础技术推导，得到了概率流常微分方程和随机微分方程两种表述形式。针对拉普拉斯分布幅度的纯跳跃过程进行了详细实现，获得了封闭解析形式的广义分数函数，并在特定参数区间内证明了其性能优于等效的高斯模型。