  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 .
