Causality--\Delta: Jacobian-Based Dependency Analysis in Flow Matching Models

TL;DR

This paper introduces a Jacobian-based analysis method for flow matching models, revealing how small perturbations propagate and uncovering dependency structures in generated features across different data domains.

Contribution

It provides a novel Jacobian-vector product approach to analyze dependencies in flow models, with closed-form solutions and empirical validation on synthetic and image data.

Findings

JVPs recover analytical Jacobians in synthetic benchmarks

Flow with attribute classifiers reveals attribute-level dependencies

Conditioning on Jacobian norms reduces correlations, indicating dependency control

Abstract

Flow matching learns a velocity field that transports a base distribution to data. We study how small latent perturbations propagate through these flows and show that Jacobian-vector products (JVPs) provide a practical lens on dependency structure in the generated features. We derive closed-form expressions for the optimal drift and its Jacobian in Gaussian and mixture-of-Gaussian settings, revealing that even globally nonlinear flows admit local affine structure. In low-dimensional synthetic benchmarks, numerical JVPs recover the analytical Jacobians. In image domains, composing the flow with an attribute classifier yields an attribute-level JVP estimator that recovers empirical correlations on MNIST and CelebA. Conditioning on small classifier-Jacobian norms reduces correlations in a way consistent with a hypothesized common-cause structure, while we emphasize that this conditioning is not a formal do intervention.