Modern successes of diffusion models in learning complex, high-dimensional
data distributions are attributed, in part, to their capability to construct
diffusion processes with analytic transition kernels and score functions. The
tractability results in a simulation-free framework with stable regression
losses, from which reversed, generative processes can be learned at scale.
However, when data is confined to a constrained set as opposed to a standard
Euclidean space, these desirable characteristics appear to be lost based on
prior attempts. In this work, we propose Mirror Diffusion Models (MDM), a new
class of diffusion models that generate data on convex constrained sets without
losing any tractability. This is achieved by learning diffusion processes in a
dual space constructed from a mirror map, which, crucially, is a standard
Euclidean space. We derive efficient computation of mirror maps for popular
constrained sets, such as simplices and $\ell_2$-balls, showing significantly
improved performance of MDM over existing methods. For safety and privacy
purposes, we also explore constrained sets as a new mechanism to embed
invisible but quantitative information (i.e., watermarks) in generated data,
for which MDM serves as a compelling approach. Our work brings new algorithmic
opportunities for learning tractable diffusion on complex domains.

Mirror Diffusion Models 是一种新的扩散模型，能在凸约束集上生成数据而不损失可解性，并且通过学习镜像映射构造的对偶空间中的扩散过程，在应用于简单 xes 和∥2 - 球等流行约束集时，MDM 展现出显著改善的性能。此外，我们还探索了约束集作为一种新机制，将隐藏但定量信息（即水印）嵌入生成的数据中，以达到安全和隐私保护的目的。我们的工作为在复杂领域中学习可解扩散提供了新的算法机会。