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A Jacobi Field Approach to Splitting Detection in Schrödinger Bridge

Chunhai Jiao, Jin Guo, Haoyan Zhang, Jinqiao Duan, Ting Gao

Abstract

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.

A Jacobi Field Approach to Splitting Detection in Schrödinger Bridge

Abstract

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.
Paper Structure (21 sections, 3 theorems, 52 equations, 16 figures, 1 algorithm)

This paper contains 21 sections, 3 theorems, 52 equations, 16 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schrödinger bridge
  4. Stochastic interpolation
  5. Problem setup
  6. Particle approximation and stochastic interpolation
  7. Jacobi field and variational dynamics
  8. Induced velocity field and Jacobian dynamics
  9. Algorithm
  10. Discontinuous Trajectory Modeling via neural networks
  11. Loss Function Designed for Physical Constraints
  12. Boundary Constraint Loss
  13. Energy Regularization
  14. Total Objective and Optimization
  15. Statistical Indicators and Top-K Analysis Framework
  16. ...and 6 more sections

Key Result

Proposition 3.1

Let $v:[0,T]\times \mathbb R^d\to \mathbb R^d$ be a sufficiently smooth velocity field, and let be the associated flow equation. Consider a smooth one parameter family of trajectories $X:(-\varepsilon,\varepsilon)\times[0,T]\to\mathbb R^d, \frac{\partial}{\partial t}X(s,t)=v\bigl(t,X(s,t)\bigr)$, here $s\in(-\varepsilon,\varepsilon)$ is a variation parameter labeling a family of nearby traject Th

Figures (16)

  • Figure 1: Probability Path Evolution. Snapshots of particle transport from a Gaussian prior to the Two Moons distribution. The $x, y$ axes represent 2D space, illustrating the structural splitting from $t=0.00$ to $t=1.00$.
  • Figure 2: Delta ($\Delta$) Distribution Evolution. Points are colored by their $\Delta$ value (splitting intensity). The $x, y$ axes are spatial coordinates. Pink circles highlight the Top-30 particles concentrated at the flow's bifurcation neck.
  • Figure 3: Max Eigenvalue Evolution. Points are colored by the Jacobian's max eigenvalue, indicating local manifold curvature. The $x, y$ axes are spatial coordinates, with pink circles tracking the Top-30 high-curvature hotspots.
  • Figure 4: Eigenvalue Indicator Analysis. (Left) Cumulative integral of Top-30 eigenvalues. (Right) Rate of Top-30 eigenvalues' change (slope). The $x$-axis is time $t$; the vertical dashed line marks the peak $t^*$ signaling the onset of splitting.
  • Figure 5: Probability Path Evolution. Snapshots of particle transport from a Gaussian prior to the Checkerboard distribution. The $x, y$ axes represent 2D space, illustrating the structural splitting from $t=0.00$ to $t=1.00$.
  • ...and 11 more figures

Theorems & Definitions (7)

  • Remark 3.1: Standard meaning of the term Jacobi field
  • Proposition 3.1
  • proof
  • Remark 3.2: Relation between the first and second order formulations
  • Proposition 3.2
  • proof
  • Proposition 3.3