针对约束生成模型的数学缺陷，提出一种低成本、灵活的分布扰动方法。
Manifold-Aware Perturbations for Constrained Generative Modeling
- 在约束条件下扰动数据分布，保留流形几何特征
- 使生成分布支持空间维度匹配环境空间，实现稳定采样
- 适用于扩散模型和归一化流，提升科学领域建模效果
生成模型在众多应用中取得成功，但在样本受等式约束的科学场景中存在固有的数学局限。本文提出一种计算成本低、数学严谨且高度灵活的分布修改方法，通过约束感知方式扰动数据分布，使新分布的支持集维度与环境空间一致，同时隐式保留底层流形结构。理论分析与多个代表性任务的实验证明，该方法能持续恢复数据分布并实现稳定采样，适用于扩散模型与归一化流。
原文摘要 · Abstract (English)
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.
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