李群流形上用于概率内在生成的薛定谔桥
Schrödinger Bridges on Lie Group Manifolds for Probabilistic Intrinsic Generation
- National Biomedical Imaging Center, College of Future Technology, Peking University（北京大学未来技术学院国家生物医学成像中心）
- State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对李群流形提出两种薛定谔桥计算实现方法，用于概率内在生成，在多个李群流形数据集上验证了方法的可行性与一致性。
AI中文摘要：
直接在几何流形上进行生成建模，可避免将非欧几里得数据扁平化、重复环境投影以及欧几里得表示中的坐标不一致性所引入的误差。薛定谔桥为规定端点分布之间的熵正则化传输提供了概率生成框架。我们研究李群流形上动力学的薛定谔桥，其状态为X_t = (g_t, ξ_t)，属于G × g，允许端点观测仅约束实际被测量的变量。具体而言，熵投影决定了未观测端点速度的条件律。针对相同的观测端点桥，我们提出两种计算实现方式：包裹核桥校准（WKBC）在紧阿贝尔群上使用显式周期动力学核；而互条件控制桥匹配（RCCBM）通过双侧端点校准和平滑化条件控制匹配处理紧非阿贝尔群。典型的教师混合路径律本身是马尔可夫互易律，因此前向生成使用校准后的初始律和一个学习到的杜布控制器。此外，我们在有界-利普希茨路径度量中建立了模块化误差界，该误差界清晰区分了由端点、控制回归、初始化、离散化及相关近似导致的误差。在多个李群流形数据集上的实验验证了我们所提方法的可行性和一致性，涵盖蛋白质和RNA扭转角、SO(3)、U(n)，以及使用紧凑降维表示的mdCATH轨迹进行的蛋白质构象转变路径生成任务。源代码可在该https URL处公开获取。
英文摘要：
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.
