2024/10/02 by Ziyu Chen, Markos A. Katsoulakis, Chen, Ziyu +3 · 1 citation
Computer Science · #FOS: Computer and information sciences #Generative Adversarial Networks and Image Synthesis #Machine Learning (cs.LG) #Machine Learning (stat.ML)
paper · pdf · doi:10.48550/arxiv.2410.01244
openalex publication_date 2024/10/02 · openalex created_date 2024/10/30 · openalex updated_date 2026/07/30
We introduce a novel Wasserstein-1 (W1) path-space divergence for stochastic and deterministic dynamics and establish a Wasserstein Uncertainty Propagation (WUP) theorem that bounds the W1 distance between terminal distributions by the proposed divergence, equivalently characterized by a weighted L2 discrepancy between the underlying drifts and the W1 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 W1 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.