Investigating noise distribution beyond Gaussian in diffusion generative models is an open problem. The Gaussian case has seen success experimentally and theoretically, fitting a unified SDE framework for score-based and denoising formulations. Recent studies suggest heavy-tailed noise distributions can address mode collapse and manage datasets with class imbalance, heavy tails, or outliers. Yoon et al. (NeurIPS 2023) introduced the L\'evy-Ito model (LIM), extending the SDE framework to heavy-tailed SDEs with $\alpha$-stable noise. Despite its theoretical elegance and performance gains, LIM's complex mathematics may limit its accessibility and broader adoption. This study takes a simpler approach by extending the denoising diffusion probabilistic model (DDPM) with $\alpha$-stable noise, creating the denoising L\'evy probabilistic model (DLPM). Using elementary proof techniques, we show DLPM reduces to running vanilla DDPM with minimal changes, allowing the use of existing implementations with minimal changes. DLPM and LIM have different training algorithms and, unlike the Gaussian case, they admit different backward processes and sampling algorithms. Our experiments demonstrate that DLPM achieves better coverage of data distribution tail, improved generation of unbalanced datasets, and faster computation times with fewer backward steps.

翻译：研究扩散生成模型中超越高斯分布的噪声分布是一个开放性问题。高斯情形在实验和理论上均取得了成功，为基于分数和去噪的公式化方法拟合了统一的随机微分方程框架。近期研究表明，重尾噪声分布能够解决模式崩溃问题，并处理具有类别不平衡、重尾或离群值的数据集。Yoon等人（NeurIPS 2023）提出了莱维-伊藤模型，将随机微分方程框架扩展至具有$\alpha$-稳定噪声的重尾随机微分方程。尽管该模型理论优雅且性能提升，但其复杂的数学处理可能限制其可及性与广泛采用。本研究采用更简化的方法，通过将去噪扩散概率模型扩展至$\alpha$-稳定噪声，构建了去噪莱维概率模型。运用基础证明技术，我们证明该模型可简化为运行标准去噪扩散概率模型，仅需极少改动即可利用现有实现。去噪莱维概率模型与莱维-伊藤模型具有不同的训练算法，且与高斯情形不同，它们允许不同的反向过程与采样算法。实验表明，去噪莱维概率模型能更好地覆盖数据分布的尾部，提升不平衡数据集的生成质量，并以更少的反向步骤实现更快的计算速度。