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Diffusion models have achieved promising results on generative learning recently. However, because diffusion processes are most naturally applied on the uncon- strained Euclidean space Rd, key challenges arise for developing diffusion based models for learning data on constrained and structured domains. We present a simple and unified framework to achieve this that can be easily adopted to various types of domains, including product spaces of any type (be it bounded/unbounded, continuous/discrete, categorical/ordinal, or their mix). In our model, the diffu- sion process is driven by a drift force that is a sum of two terms: one singular force designed by Doob’s h-transform that ensures all outcomes of the process to belong to the desirable domain, and one non-singular neural force field that is trained to make sure the outcome follows the data distribution statistically. Ex- periments show that our methods perform superbly on generating tabular data, images, semantic segments and 3D point clouds. Code is available at https: //github.com/gnobitab/ConstrainedDiffusionBridge. 

We propose a family of First Hitting Diffusion Models (FHDM), deep generative models that generate data with a diffusion process that terminates at a random first hitting time. This yields an extension of the standard fixed-time diffusion models that terminate at a pre-specified deterministic time. Although standard diffusion models are designed for continuous unconstrained data, FHDM is natu- rally designed to learn distributions on continuous as well as a range of discrete and structure domains. Moreover, FHDM enables instance-dependent terminate time and accelerates the diffusion process to sample higher quality data with fewer diffusion steps. Technically, we train FHDM by maximum likelihood estimation on diffusion trajectories augmented from observed data with conditional first hitting processes (i.e., bridge) derived based on Doob’s h-transform, deviating from the commonly used time-reversal mechanism. We apply FHDM to generate data in various domains such as point cloud (general continuous distribution), climate and geographical events on earth (continuous distribution on the sphere), unweighted graphs (distribution of binary matrices), and segmentation maps of 2D images (high-dimensional categorical distribution). We observe considerable improvement compared with the state-of-the-art approaches in both quality and speed.