Point processes model the distribution of random point sets in mathematical spaces, such as spatial and temporal domains, with applications in fields like seismology, neuroscience, and economics. Existing statistical and machine learning models for point processes are predominantly constrained by their reliance on the characteristic intensity function, introducing an inherent trade-off between efficiency and flexibility. In this paper, we introduce Point Set Diffusion, a diffusion-based latent variable model that can represent arbitrary point processes on general metric spaces without relying on the intensity function. By directly learning to stochastically interpolate between noise and data point sets, our approach enables efficient, parallel sampling and flexible generation for complex conditional tasks defined on the metric space. Experiments on synthetic and real-world datasets demonstrate that Point Set Diffusion achieves state-of-the-art performance in unconditional and conditional generation of spatial and spatiotemporal point processes while providing up to orders of magnitude faster sampling than autoregressive baselines.

翻译：点过程模拟随机点集在数学空间（如空间域和时间域）中的分布，其应用领域涵盖地震学、神经科学和经济学等。现有的点过程统计与机器学习模型主要受限于对特征强度函数的依赖，这导致效率与灵活性之间存在固有的权衡。本文提出点集扩散（Point Set Diffusion），这是一种基于扩散的隐变量模型，能够在一般度量空间上表示任意点过程，而无需依赖强度函数。通过直接学习在噪声与数据点集之间进行随机插值，我们的方法能够为定义在度量空间上的复杂条件任务实现高效、并行的采样与灵活生成。在合成数据集和真实数据集上的实验表明，点集扩散在空间与时空点过程的无条件及条件生成任务中达到了最先进的性能，同时其采样速度比自回归基线方法快数个数量级。