Generative models have enjoyed widespread success in a variety of applications. However, they encounter inherent mathematical limitations in modeling distributions where samples are constrained by equalities, as is frequently the setting in scientific domains. In this work, we develop a computationally cheap, mathematically justified, and highly flexible distributional modification for combating known pitfalls in equality-constrained generative models. We propose perturbing the data distribution in a constraint-aware way such that the new distribution has support matching the ambient space dimension while still implicitly incorporating underlying manifold geometry. Through theoretical analyses and empirical evidence on several representative tasks, we illustrate that our approach consistently enables data distribution recovery and stable sampling with both diffusion models and normalizing flows.

翻译：生成模型已在多种应用中取得了广泛成功。然而，在样本受等式约束的分布建模中，它们面临着固有的数学局限性，而这正是科学领域中常见的设定。本研究提出了一种计算成本低廉、数学依据充分且高度灵活的分布修正方法，以应对等式约束生成模型中已知的缺陷。我们建议以约束感知的方式扰动数据分布，使得新分布的支持集与外围空间维度相匹配，同时仍能隐式地融入底层流形几何结构。通过理论分析和多个代表性任务的实证证据，我们证明该方法能够持续实现数据分布恢复，并在扩散模型和归一化流中实现稳定采样。