Self-Consistent Generative Paths via Admissible Random Variational Transport

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.

The slow iterative sampling nature remains a major bottleneck for the practical deployment of diffusion and flow-based generative models. While consistency models (CMs) represent a state-of-the-art distillation-based approach for efficient generation, their large-scale application is still limited by two key issues: training instability and inflexible sampling. Existing methods seek to mitigate these problems through architectural adjustments or regularized objectives, yet overlook the critical reliance on trajectory selection. In this work, we first conduct an analysis on these two limitations: training instability originates from loss divergence induced by unstable self-supervised term, whereas sampling inflexibility arises from error accumulation. Based on these insights and analysis, we propose the Dual-End Consistency Model (DE-CM) that selects vital sub-trajectory clusters to achieve stable and effective training. DE-CM decomposes the PF-ODE trajectory and selects three critical sub-trajectories as optimization targets. Specifically, our approach leverages continuous-time CMs objectives to achieve few-step distillation and utilizes flow matching as a boundary regularizer to stabilize the training process. Furthermore, we propose a novel noise-to-noisy (N2N) mapping that can map noise to any point, thereby alleviating the error accumulation in the first step. Extensive experimental results show the effectiveness of our method: it achieves a state-of-the-art FID score of 1.70 in one-step generation on the ImageNet 256x256 dataset, outperforming existing CM-based one-step approaches.

Modern generative models are increasingly trained using model-generated signals, creating both opportunities for self-improvement and risks of collapse. We study optimal self-distillation (SD) for rectified flow (RF): given a suboptimal teacher velocity field, can a student trained on a mixture of true RF velocities and teacher velocities provably improve the teacher? For linear RF with ridge regularization on fixed interpolation pairs, we prove an exact affine path identity, derive the optimal mixing coefficient in closed form, and show strict improvement in integrated velocity risk whenever the teacher risk is nonstationary along the regularization path. The optimal coefficient obeys a sign rule: positive mixing corrects under-regularized teachers, while negative mixing corrects over-regularized teachers. We also give one-shot generalized cross-validation (GCV) and validation tuning procedure that avoids grid search over mixing weights and repeated refitting. Combining this theorem with RF Wasserstein convergence bounds, we show that optimal self-distillation improves the velocity estimation terms controlling continuous-time and finite-step generation error. Experiments with Gaussian models, Gaussian mixtures, and image data show that optimal self-distillation improves velocity risk, mode recovery, and finite-step generation relative to both the teacher and pure distillation.