Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergences

We introduce a novel Wasserstein-1 ($W_1$) path-space divergence for stochastic and deterministic dynamics and establish a Wasserstein Uncertainty Propagation (WUP) theorem that bounds the $W_1$ distance between terminal distributions by the proposed divergence, equivalently characterized by a weighted $L^2$ discrepancy between the underlying drifts and the $W_1$ distance between their initial measures.

A key ingredient is a probabilistic framework combining adjoint Feynman-Kac representations with synchronous coupling (and reflection coupling on bounded domains), yielding Wasserstein stability estimates beyond existing PDE- and Girsanov-based approaches.

The framework accommodates time-varying and possibly degenerate diffusion coefficients, empirical and singular measures, and remains valid in the deterministic limit of flow matching.

Unlike KL-based uncertainty quantification bounds, it does not require absolute continuity of path measures and therefore remains well-defined in singular settings.

As consequences of the WUP theorem, we derive $W_1$ robustness and generalization bounds for score-based generative models and flow matching at both population and finite-sample levels.

We further specialize the framework to group-symmetric targets, providing the first error analysis of equivariant flow-based models and the first quantitative comparison between data augmentation and equivariant inductive bias.

Our analysis identifies a symmetry-aware Wasserstein path-space divergence that quantifies the model-form error induced by non-equivariant parametrizations.

We prove that this error cannot be removed by additional data or training and vanishes only under equivariant architectures, establishing a precise theoretical advantage of equivariant inductive bias over data augmentation.

Numerical experiments on group-symmetric Gaussian mixtures corroborate the theory.