Diffusion Generative Modeling on Lie Group Representations

TL;DR

This paper introduces a new class of diffusion processes operating directly on Lie group representations, enabling more effective modeling of complex data distributions on non-Abelian groups with applications in molecular conformer generation and docking.

Contribution

It develops a generalized score-matching framework for Lie groups, deriving Langevin dynamics and stochastic differential equations tailored for non-Abelian Lie group representations.

Findings

Effective modeling of molecular conformers using SO(3)

Improved ligand-specific transformations over Riemannian diffusion

Reduced dimensionality enhances learning efficiency

Abstract

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.