First Hitting Diffusion Models for Generating Manifold, Graph and Categorical Data
This work addresses the challenge of generating manifold, graph, and categorical data, which is important for applications in fields like climate science and image segmentation, though it appears incremental as an extension of standard diffusion models.
The authors tackled the problem of generating data across continuous, discrete, and structured domains by proposing First Hitting Diffusion Models (FHDM), which use a diffusion process with random termination times, resulting in considerable improvements in quality and speed compared to state-of-the-art methods.
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 naturally 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.