Denoising Diffusion Probabilistic Models (DDPMs) provide the foundation for the recent breakthroughs in generative modeling. Their Markovian structure makes it difficult to define DDPMs with distributions other than Gaussian or discrete. In this paper, we introduce Star-Shaped DDPM (SS-DDPM). Its star-shaped diffusion process allows us to bypass the need to define the transition probabilities or compute posteriors. We establish duality between star-shaped and specific Markovian diffusions for the exponential family of distributions and derive efficient algorithms for training and sampling from SS-DDPMs. In the case of Gaussian distributions, SS-DDPM is equivalent to DDPM. However, SS-DDPMs provide a simple recipe for designing diffusion models with distributions such as Beta, von Mises$\unicode{x2013}$Fisher, Dirichlet, Wishart and others, which can be especially useful when data lies on a constrained manifold. We evaluate the model in different settings and find it competitive even on image data, where Beta SS-DDPM achieves results comparable to a Gaussian DDPM. Our implementation is available at https://github.com/andrey-okhotin/star-shaped .

翻译：去噪扩散概率模型（DDPMs）为生成建模领域的最新突破奠定了基础。由于其马尔可夫链结构，定义非高斯或离散分布的DDPMs存在困难。本文提出星形去噪扩散概率模型（SS-DDPM）。其星形扩散过程使我们无需定义转移概率或计算后验分布。我们建立了指数族分布中星形扩散与特定马尔可夫扩散之间的对偶性，并推导了SS-DDPMs的高效训练与采样算法。在高斯分布情形下，SS-DDPM与DDPM等价。然而，SS-DDPM为设计基于贝塔分布、冯·米塞斯-费舍尔分布、狄利克雷分布、威沙特分布等分布的扩散模型提供了简洁方法，尤其适用于数据位于约束流形的情况。我们在不同设置下评估该模型，发现即使在图像数据上，贝塔SS-DDPM也能取得与高斯DDPM相当的结果。我们的实现已开源至 https://github.com/andrey-okhotin/star-shaped 。