  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.
