Generative Modeling on Lie Groups via Euclidean Generalized Score Matching

2025-02-04

AI Summary

This work addresses the challenge of generative modeling on Lie groups—particularly non-Abelian ones such as SO(3) and SE(3)—by introducing the first score-based diffusion framework built upon paired stochastic differential equations (SDEs). Methodologically, it generalizes fractional score matching to Lie group manifolds via tangent-space parameterization in the associated Lie algebra, enabling equivariant Langevin dynamics in Euclidean space and circumventing the geometric complexity of solving SDEs directly on the manifold; it unifies standard score matching and flow matching. Key contributions include the first unbiased score estimation and generative sampling on arbitrary Lie groups, overcoming functional-space limitations inherent in prior equivariant models. Experiments demonstrate significant improvements over Riemannian diffusion baselines on SO(3)-based molecular conformation generation and SE(3)-based molecular docking, achieving reduced trajectory dimensionality, enhanced learning efficiency, and support for cross-distribution transfer modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

We extend Euclidean score-based diffusion processes to generative modeling on Lie groups. Through the formalism of Generalized Score Matching, our approach yields a Langevin dynamics which decomposes as a direct sum of Lie algebra representations, enabling generative processes on Lie groups while operating in Euclidean space. Unlike equivariant models, which restrict the space of learnable functions by quotienting out group orbits, our method can model 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. Additionally, we demonstrate the universality of our approach by deriving how it extends to flow matching.

Problem

• Generative modeling on Lie groups
• Generalized Score Matching formalism
• Euclidean space Langevin dynamics

Innovation

• Generalized Score Matching
• Lie group generative modeling
• paired SDEs