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Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups
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Proposes Trivialized Diffusion Model (TDM), which runs a trivialized kinetic Langevin noising process on compact Lie groups so the score is learned in a fixed Euclidean Lie algebra space, with an operator-splitting manifold-preserving sampler.
Problem
Generative modeling of data on Lie groups (torus, SO(n), U(n), protein/RNA torsion angles) where prior Riemannian diffusion methods need tangent-space projections and retractions that introduce approximation errors.
Approach
Introduce a left-trivialized momentum variable in the Lie algebra so that the forward kinetic Langevin process on G x g has no direct noise on g; the only score needed is in the fixed vector space g, and an operator-splitting integrator keeps samples exactly on the group without projections.
Key results
- On protein/RNA torsion datasets TDM reports test NLL of 0.69 on General (2D) vs 1.01 for RFM, and -6.86 on RNA (7D) vs -5.20 for RFM.
- TDM reports NLL -0.60 on Proline (2D) vs 0.15 for RFM and 0.52 on Pre-Pro vs 1.18 for RFM.
- Claims first results generating SO(n) for n>3 and U(n) quantum-evolution data, where no prior baseline is reported.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use TDM instead of RFM | Protein General (2D) · Protein General (2D) test | NLL: 1.01 \pm 0.025 -> 0.69 \pm 0.14 | Helped |
| Use TDM instead of RFM | Protein Glycine (2D) · Protein Glycine (2D) test | NLL: 1.90 \pm 0.055 -> 1.04 \pm 0.27 | Helped |
| Use TDM instead of RDM | Protein Proline (2D) · Protein Proline (2D) test | NLL: 0.12 \pm 0.011 -> -0.60 \pm 0.15 | Helped |
| Use TDM instead of RFM | Protein Pre-Pro (2D) · Protein Pre-Pro (2D) test | NLL: 1.18 \pm 0.055 -> 0.52 \pm 0.10 | Helped |
| Use TDM instead of RFM | RNA (7D) · RNA (7D) test | NLL: -5.20 \pm 0.067 -> -6.86 \pm 0.46 | Helped |
Limitations
- Trivialization relies on the group structure and does not directly generalize to general manifolds (only possibly homogeneous spaces).
- Conditional transition probability closed form only for abelian groups (tori/SO(2) products); non-abelian cases use implicit score matching with Hutchinson trace estimation.
Experimental setup
Datasets: checkerboard on T^2 Pacman on T^2 protein torsion angle datasets (General, Glycine, Proline, Pre-Pro) RNA torsion angle dataset (7D) SO(n) synthetic mixture distributions quantum oscillator and transverse field Ising unitary time-evolution operators
Benchmarks: checkerboard T^2 Pacman T^2 protein torsion angles (General, Glycine, Proline, Pre-Pro) RNA torsion angles (7D) SO(3)/SO(n) mixtures U(n) quantum ensembles
Metrics: negative log-likelihood
Baselines: RFM RDM RSGM FrameDiff
Builds on: Riemannian Flow Matching Riemannian Score-based Model Riemannian Diffusion Model Critically-damped Langevin diffusion (CLD)
Closest papers by meaning
- Diffusion Generative Modeling on Lie Group Representations (NeurIPS 2025 · similarity 0.87)
- Efficient Diffusion Models for Symmetric Manifolds (ICML 2025 · similarity 0.86)
- Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes (ICML 2024 · similarity 0.85)
- Rao-Blackwell Gradient Estimators for Equivariant Denoising Diffusion (NeurIPS 2025 · similarity 0.85)
- Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion (ICLR 2025 · similarity 0.85)
- SymDrift: One-Shot Generative Modeling under Symmetries (NeurIPS 2026 · similarity 0.84)
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