在李群表示空间构建扩散模型，提升分子构型生成效率
Diffusion Generative Modeling on Lie Group Representations
- 将扩散模型扩展到李群表示空间，通过李代数分解实现复杂分布建模
- 在SO(3)和SE(3)上生成分子构象与对接变换，优于传统黎曼扩散方法
- 适合需要几何结构先验的分子生成、3D形状建模等任务
我们提出一类新型基于得分的扩散过程，直接在李群的表示空间中运行。利用广义得分匹配框架，推导出一类可分解为李代数表示直和的朗之万动力学，从而实现对任意非阿贝尔李群上目标分布的建模。当李群为平移群时，标准得分匹配为其特例。我们证明了该广义生成过程是首次提出的成对随机微分方程（SDEs）的解。通过多类数据实验验证，该方法在SO(3)引导的分子构象生成和配体特异性全局SE(3)变换建模中表现优异，相比在群本身上的黎曼扩散有明显提升。恰当选择李群可降低轨迹空间的有效维度，提高学习效率，并支持复杂数据分布间的转换建模。
原文摘要 · Abstract (English)
We introduce a novel class of score-based diffusion processes that operate directly in the representation space of Lie groups. Leveraging the framework of Generalized Score Matching, we derive a class of Langevin dynamics that decomposes as a direct sum of Lie algebra representations, enabling the modeling of any target distribution on any (non-Abelian) Lie group. Standard score-matching emerges as a special case of our framework when the Lie group is the translation group. We prove that our generalized generative processes arise as solutions to a new class of paired stochastic differential equations (SDEs), introduced here for the first time. We validate our approach through experiments on diverse data types, demonstrating its effectiveness in real-world applications such as SO(3)-guided molecular conformer generation and modeling ligand-specific global SE(3) transformations for molecular docking, showing improvement in comparison to Riemannian diffusion on the group itself. We show that an appropriate choice of Lie group enhances learning efficiency by reducing the effective dimensionality of the trajectory space and enables the modeling of transitions between complex data distributions.
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