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Diffusion Generative Modeling on Lie Group Representations
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Proposes score-based diffusion directly in the representation space of Lie groups via generalized score matching, with paired SDEs whose Langevin dynamics decompose over Lie algebra infinitesimal actions, and validates on 2D/3D/4D toy data, rotated MNIST, QM9 conformers and CrossDocked SE(3) docking.
Problem
Diffusion on curved Lie group manifolds is hard to parametrize and simulate (projections, non-Abelian groups), and existing approaches often perform worse than unconstrained Cartesian diffusion; the paper asks how to build a generative process on the flat space X that models any distribution on a Li
Approach
Use generalized score matching with the linear operator given by fundamental vector fields of the Lie algebra acting on Euclidean space, yielding a new class of solvable paired SDEs (with a quadratic Casimir correction term) whose reverse process is simulation-free and stays in the flat representation space.
Key results
- On rotated MNIST, the GSM model reaches 0.96 classification accuracy and FID 85.77 versus 0.80 and 133.4 for BBDM.
- On CrossDocked2020 docking, the Lie algebra method reaches 2.91 ± 1.0 Å RMSD versus 5.6 ± 1.2 Å for RSGM, and is comparable to BBDM (2.92 ± 1.57 Å).
- On QM9, the SO(3)xR+ model yields mean UFF energy difference -0.2159 versus -0.2144 for standard diffusion, a marginal gap.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Lie group representation diffusion (GSM) instead of BBDM | MNIST rotation bridging · Rotated MNIST | classification accuracy: 0.80 vs 0.96 | Helped |
| Use Lie group representation diffusion (GSM) instead of BBDM | MNIST rotation bridging · Rotated MNIST | FID: 133.4 vs 85.77 | Helped |
| Use SO(3) × R₊ dynamics instead of standard translation diffusion | QM9 conformer generation · QM9 | mean Δ energy: -0.2144 -> -0.2159 | Helped |
| Use Lie algebra-guided diffusion instead of RSGM | Molecular docking · CrossDocked2020 | RMSD (Å): 5.6± 1.2 -> 2.9± 1.0 | Helped |
| Use Lie algebra-guided diffusion instead of BBDM | Molecular docking · CrossDocked2020 test | RMSD (Å): 2.92 ± 1.57 vs 2.91 ± 1.0 | No clear effect |
| Constrain the generated trajectory to global SO(3) rotations | Molecular docking · CrossDocked2020 test | MAE(D(x₀, x₀), D(x̂₀, x̂₀)): 0.43± 0.21 -> 0.0 | Helped |
Surprising findings
- The standard BBDM baseline on CrossDocked violates the global SO(3) symmetry structure, giving MAE 0.43 ± 0.21 on the self-consistency check while the Lie-algebra methods achieve 0.0 by design.
- Non-commuting Lie algebra matrices can still yield commuting differential operators on X, so non-Abelian groups can be used despite non-commuting generators.
Limitations
- Does not extend to finite groups.
- Main theoretical results restricted to actions on vector spaces X rather than general manifolds.
- QM9 improvement over standard diffusion is marginal.
Experimental setup
Datasets: MNIST QM9 CrossDocked2020
Benchmarks: rotated MNIST QM9 CrossDocked2020
Metrics: classification accuracy FID RMSD UFF energy difference W2 distance
Baselines: Brownian Bridge Diffusion Model (BBDM) RSGM standard Euclidean denoising score matching diffusion
Builds on: Generalized Score Matching score-based SDEs Brownian Bridge Diffusion Model Riemannian score-based generative modeling (RSGM)
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