Sampling viable 3D structures (e.g., molecules and point clouds) with
SE(3)-invariance using diffusion-based models proved promising in a variety of
real-world applications, wherein SE(3)-invariant properties can be naturally
characterized by the inter-point distance manifold. However, due to the
non-trivial geometry, we still lack a comprehensive understanding of the
diffusion mechanism within such SE(3)-invariant space. This study addresses
this gap by mathematically delineating the diffusion mechanism under
SE(3)-invariance, via zooming into the interaction behavior between coordinates
and the inter-point distance manifold through the lens of differential
geometry. Upon this analysis, we propose accurate and projection-free diffusion
SDE and ODE accordingly. Such formulations enable enhancing the performance and
the speed of generation pathways; meanwhile offering valuable insights into
other systems incorporating SE(3)-invariance.

通过微分几何的视角，本研究对 SE (3) 不变空间中的扩散机制进行了数学界定，揭示了坐标与点间距流形之间的相互作用行为，并提出了准确且无需投影的扩散 SDE 和 ODE 公式，从而提高了生成路径的性能和速度，并为其他包含 SE (3) 不变性的系统提供了有价值的见解。