Causality--Δ: Jacobian-Based Dependency Analysis in Flow Matching Models

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.

💡 Research Summary

This paper investigates the internal dependency structure of flow‑matching generative models by exploiting Jacobian‑vector products (JVPs) as a practical tool for causal‑style analysis. Flow matching learns a time‑dependent velocity field vₜ(x) that deterministically transports a simple base distribution (typically a standard Gaussian) into a target data distribution via the ODE ϕₜ. The authors first derive closed‑form expressions for the optimal drift u(xₜ,t)=E