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This paper explores the application of Schr枚dinger bridges for generative modeling on Lie group manifolds, addressing the limitations of traditional Euclidean representations in capturing non-Euclidean data. By developing two computational methods鈥擶rapped-Kernel Bridge Calibration (WKBC) for compact Abelian groups and Reciprocal Conditional-Control Bridge Matching (RCCBM) for compact non-Abelian groups鈥攖he authors demonstrate how to effectively generate probabilistic paths constrained by observed endpoints. Experimental results across various datasets, including protein and RNA torsions, confirm the methods' feasibility and highlight a modular error bound that clarifies the sources of approximation errors.
Generative modeling on Lie group manifolds reveals that traditional Euclidean approaches can obscure critical dynamics, leading to significant errors in data representation.
Generative modeling directly on geometric manifolds can avoid errors introduced by flattening non-Euclidean data, repeated ambient projection, and coordinate inconsistency in Euclidean representations. Schrodinger bridges provide a probabilistic generative framework for entropy-regularized transport between prescribed endpoint distributions. We study Schrodinger bridges for kinetic dynamics on Lie group manifolds with state X_t = (g_t, xi_t) in G x g, allowing endpoint observations to constrain only the variables that are actually measured. In particular, the entropy projection determines the conditional law of the unobserved endpoint velocities. For the same observed endpoint bridge, we develop two computational realizations: Wrapped-Kernel Bridge Calibration (WKBC) uses an explicit periodized kinetic kernel on compact Abelian groups, whereas Reciprocal Conditional-Control Bridge Matching (RCCBM) handles compact non-Abelian groups through two-sided endpoint calibration and mollified conditional-control matching. The canonical teacher-mixture path law is itself a Markov reciprocal law, so forward generation uses a calibrated initial law and one learned Doob controller. Moreover, we establish a modular error bound in the bounded-Lipschitz path metric that provides a clean separation of errors due to endpoints, control regression, initialization, discretization, and related approximations. Experiments on multiple Lie group manifold datasets validate the feasibility and consistency of our proposed method, covering protein and RNA torsions, SO(3), U(n), and the Protein Conformational Transition Pathway Generation task using mdCATH trajectories in a compact reduced representation. The source code is publicly available at https://github.com/cafferyzhang12/Schr-dinger_Bridge_on_LieGroup.
