Decomposing Out-of-Distribution Error in Conditional Flow Matching via Wasserstein Geometry

Conditional flow matching has emerged as a powerful generative modeling framework that learns a vector field to transport an initial distribution toward a target data distribution. However, theoretical understanding of its out-of-distribution (OOD) performance under unseen conditions remains limited. In this work, we establish a rigorous geometric formulation to decompose the source of generalization error. We treat the conditional task as a map from the condition space to the Wasserstein space and derive a generalization bound under a coarse embedding assumption. The resulting decomposition separates OOD error into three tractable components: Interpolation Sparsity, Geometric Distortion, and In-Distribution Fit. Our empirical evaluation confirms that this framework demonstrates three key functions: (1) it acts as a diagnostic tool that tracks the dynamics of generalization during training; (2) it identifies dataset-specific failure modes (e.g., topological gaps, geometric instability); and (3) it enables mathematically motivated interventions that yield predictable gains by minimizing specific terms.