基于雅可比场的随机桥中路径分裂检测方法
A Jacobi Field Approach to Splitting Detection in Schrödinger Bridge
AI总结：
本文提出基于雅可比场的指标，用于检测随机桥中路径分裂的起始点和位置，通过分析雅可比场的谱结构量化扰动放大，适用于非凸和不连通的目标分布。
摘要：
我们研究了在概率分布之间进行随机插值时检测路径分裂起始点的问题。当目标分布非凸或支持在不连通组件上时，插值轨迹可能分裂成不同分支。受随机控制和随机桥观点启发，我们提出一个基于雅可比场的指标来识别候选分裂时间和位置。我们的方法基于诱导插值流的线性化雅可比场。从随机插值假设出发，通过条件平均构建欧拉速度场，并推导其空间雅可比在目标样本云局部后验几何中的表达。这使我们能够将雅可比的对称部分解释为局部应变张量，并利用其谱结构量化沿参考轨迹的微小扰动放大。在非凸和不连通目标分布上的数值实验表明，所提指标能一致定位分支区域的出现并捕捉分裂的时间发展。这些结果表明，雅可比场分析为研究随机插值中的局部不稳定性及分裂现象提供了自然的数学框架。
英文摘要：
We study the problem of detecting the onset of path splitting in stochastic interpolation between probability distributions. This question is especially subtle when the target distribution is nonconvex or supported on disconnected components, where interpolating trajectories may separate into distinct branches. Motivated by the stochastic control and Schrödinger bridge viewpoint, we propose a Jacobi field based indicator for identifying candidate splitting times and locations. Our approach is based on the Jacobi field associated with the linearization of an induced interpolating flow. Starting from a stochastic interpolation ansatz, we construct an Eulerian velocity field by conditional averaging and derive its spatial Jacobian in terms of the local posterior geometry of the target sample cloud. This allows us to interpret the symmetric part of the Jacobian as a local strain tensor and to use its spectral structure to quantify the amplification of infinitesimal perturbations along reference trajectories. Numerical experiments on non-convex and disconnected target distributions show that the proposed indicator consistently localizes the emergence of branching regions and captures the temporal development of splitting. These results suggest that Jacobi field analysis provides a natural mathematical framework for studying local instability and splitting phenomena in stochastic interpolation.
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