Trivialized Generative Models on Lie Groups

Many problems in diverse fields involve data that naturally live on Lie groups. However, existing Riemannian and Lie group generative models still face several limitations. Riemannian flow and consistency models often require position-dependent vector fields, and expensive geometric terms such as Euler-Arnold simulation on general Lie groups or covariant derivatives. Existing Lie group flow models are usually tied to collinear exponential paths, while momentum-based Lie group diffusion relies on compactness assumptions and does not naturally extend to non-compact groups. To address these limitations, we propose Trivialized Generative Models (TGM), a family of generative models that learns endpoint-constrained paths in the fixed Lie algebra and lifts them to the Lie group. This yields Trivialized Flow Matching (TFM), Trivialized Consistency Models (TCM), and momentum-based extensions with endpoint correction, enabling flexible path design, simpler few-step objectives, and generation on non-compact Lie groups. We evaluate TGM on compact SO(3) benchmarks, non-compact $\mathrm{Sp}(4,\mathbb{R})$ datasets motivated by continuous quantum systems and paraxial optics, and real-world protein backbone generation. Across these settings, TGM improves sample quality and efficiency over existing Riemannian and Lie group baselines.