Schrödinger Bridges on Lie Group Manifolds for Probabilistic Intrinsic Generation

2026-09-02

AI Summary

该研究使用李群流形上的薛定谔桥解决几何流形上概率生成问题，提出两种计算方法：WKBC和RCCBM，并在多个数据集上验证了方法的有效性。

Abstract

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.

Problem

• Schrödinger Bridges
• Lie Group Manifolds
• Generative Modeling
• Non-Euclidean Data
• Endpoint Distributions

Innovation

• Schrödinger Bridges
• Lie Group Manifolds
• Probabilistic Generation
• Wrapped-Kernel Bridge Calibration (WKBC)
• Reciprocal Conditional-Control Bridge Matching (RCCBM)