A Jacobi Field Approach to Splitting Detection in Schr\"{o}dinger Bridge

TL;DR

This paper introduces a Jacobi field-based method to detect the onset of path splitting in stochastic interpolation, especially for complex target distributions, by analyzing local instability through the spectral properties of the Jacobian.

Contribution

It proposes a novel Jacobi field indicator derived from the linearization of the interpolating flow to identify splitting points in Schrödinger bridge problems.

Findings

The indicator effectively localizes splitting regions in non-convex distributions.

It captures the temporal development of branching in stochastic interpolations.

Numerical experiments validate the approach on complex target distributions.

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 Schr\"odinger 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.