Lie Group-Induced Dynamics in Score-Based Generative Modeling
Keywords: Score-matching, Generative modeling, Denoising Diffusion models, Lie groups, Lie algebras, Molecular Conformer Generation, Deep Learning
TL;DR: We generalize generative diffusion processes by applying generalized score-matching to Lie algebra representations, resulting in an interpretable sampling denoising dynamics that follows the orbits of any Lie group acting on the data space.
Abstract: We extend score-based generative modeling by incorporating Lie group actions on the data manifold into the denoising diffusion process. Our approach yields a Langevin dynamics whose infinitesimal transformations decompose as a direct sum of Lie algebra representations, enabling generative processes that align with the underlying symmetry properties of the data. Unlike equivariant models, which restrict the space of learnable functions by quotienting out group orbits, our method incorporates both global and local symmetries and can model any target distribution. Standard score-matching, which minimizes the Fisher divergence, emerges as a special case of our framework when the Lie group is the translation group in Euclidean space. We prove that our generalized generative processes arise as solutions to a new class of reverse-time 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. 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, lifting the requirement of a Gaussian prior. Additionally, we demonstrate the universality of our approach by deriving how it extends to flow matching techniques.
Supplementary Material: zip
Primary Area: generative models
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Submission Number: 12160
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