flow matching

This work addresses the theoretical gap in sample complexity analysis for flow-matching generative models. Unlike prior studies relying on empirical risk minimization (ERM) assumptions, we establish the first end-to-end upper bound on sample complexity without such assumptions. Methodologically, we model the continuous flow via ordinary differential equations and parameterize the velocity field using neural networks; we then introduce a triple-error decomposition framework—comprising neural approximation error, statistical error, and optimization error—and rigorously analyze its convergence. Our theoretical analysis shows that $O(varepsilon^{-4})$ samples suffice to achieve $O(varepsilon)$ generative accuracy in the Wasserstein-2 distance. This constitutes the first rigorous, non-ERM-dependent sample complexity guarantee for flow matching, filling a critical theoretical void. Moreover, our result provides foundational insights for efficient training and generalization analysis of flow-based generative models.

This work addresses the modeling and learning of velocity fields governing data distribution evolution in flow matching, aiming to unify transport planning, Markov kernels, and stochastic process paradigms. Methodologically, it establishes the first theoretical equivalence framework for velocity fields characterizing absolutely continuous Wasserstein curves across these three constructions; introduces the conditional Wasserstein distance as a novel metric for Bayesian inverse problems; and unifies the geometric interpretations of continuous normalizing flows and score matching. Leveraging tools from Wasserstein geometry, optimal transport, and stochastic differential equations, the paper rigorously proves the intrinsic consistency of multiple velocity field learning approaches, thereby strengthening the mathematical foundations of flow matching. Experiments demonstrate that the proposed framework effectively generates high-fidelity conditional distributions in Bayesian inverse problems.

This work proposes a continuous-time generative modeling framework inspired by the Chow–Rashevskii theorem to address the poor parameter efficiency of conventional generative models in high-dimensional spaces, which often struggle to balance expressiveness and scalability. By composing a small set of fixed vector fields with learnable scalar control functions, the method constructs transport maps on manifolds, decoupling model parameters into low-dimensional scalar control channels. This design significantly enhances both parameter efficiency and interpretability. Leveraging Lie algebra–based controllable vector fields, continuous normalizing flows, and ordinary differential equation solvers, the framework enables efficient distribution transformation. Experiments demonstrate that the model accurately fits complex synthetic distributions using only a minimal number of control channels, confirming its strong representational capacity and computational efficiency.

Generative models such as diffusion models rely on multi-step numerical integration for inference, incurring high computational cost; while consistency models enable one-step generation, they lack a unified theoretical foundation. Method: We propose a novel paradigm that directly learns the flow map between two time points of an ordinary differential equation (ODE), unifying consistency modeling, progressive distillation, and other few-step generation approaches. Leveraging stochastic interpolation, we jointly optimize flow map prediction and velocity field distillation loss, integrating ODE theory with neural operator principles to enable tunable step counts for precision–efficiency trade-offs. Contribution/Results: On CIFAR-10 and ImageNet 32×32, our method achieves 10–50× speedup in sampling over standard diffusion models while preserving competitive sample quality—demonstrating both theoretical coherence and practical efficacy in accelerating generative inference.

This work addresses the computational challenge of dynamic optimal transport (OT) between high-dimensional sample distributions. We propose an end-to-end modeling framework based on invertible flow neural networks. Unlike conventional OT solvers that rely on density estimation or discretization, our method directly learns an invertible transport map from finite samples to minimize the Wasserstein-2 cost, while enforcing dynamical consistency via ordinary differential equation (ODE) constraints. To our knowledge, this is the first approach to employ normalizing flows for dynamic OT between arbitrary empirical distributions. It enables infinitesimal density ratio estimation and continuous latent-space interpolation. Our method achieves state-of-the-art performance on high-dimensional density ratio estimation, standard OT benchmarks, and image translation tasks—outperforming existing approaches in accuracy and robustness. Theoretical analysis ensures rigorous OT compliance, and empirical evaluation confirms scalability to high-dimensional settings, bridging theoretical soundness with practical applicability.

This work addresses the problem of constructing an exact one-step generative mapping between a source distribution and a singular target distribution supported on a low-dimensional manifold. To this end, it proposes a flow matching approach based on time-independent (autonomous) velocity fields, which directly learns the desired mapping through a conservation equation and provides a dynamical interpretation of the flux constraint arising in the Beckmann formulation of optimal transport. Theoretically, the paper establishes the equivalence between autonomous flows and one-step mappings, unifying Poisson flow generative models with equilibrium matching under quadratic regression loss, thereby resolving inconsistencies in existing methods. Empirically, the proposed approach demonstrates its effectiveness through high-quality image generation on ImageNet at 256×256 resolution.

This work addresses the limitation of existing flow matching methods, which fail to guarantee path independence under multi-parameter variations, leading to generation processes that depend on specific transport trajectories. To overcome this, the paper proposes Path-independent Flow Matching (PiFM), which learns vector fields satisfying path-independent differential constraints, thereby extending flow matching to high-dimensional parameter spaces and enabling consistent, composable transport determined solely by initial and terminal distributions. Theoretical analysis reveals an intrinsic connection between PiFM and Wasserstein barycenters, and a computationally tractable training objective is derived via multi-parameter conditional probability path regression without requiring simulation. Experiments demonstrate that PiFM outperforms current approaches on both synthetic and real-world data, generating path-independent trajectories and effectively synthesizing high-quality out-of-distribution samples.

This work investigates the statistical limits and sample complexity lower bounds for estimating any valid transport map without relying on optimal transport (OT). Formalizing the estimation task within a minimax framework, the study integrates tools from optimal transport theory, minimax analysis, and stability-based hypothesis testing to establish, for the first time, rigorous statistical lower bounds for non-optimal yet valid transport maps. The results demonstrate that under standard stability conditions, the statistical difficulty of estimating an arbitrary valid transport map is comparable to that of estimating the OT map itself. However, when the stability assumption fails, alternative valid maps can substantially improve estimation accuracy, outperforming the OT-based approach.

This study addresses the challenge of learning the time-evolving probability density dynamics of high-dimensional systems from unlabeled snapshot data alone, without access to true trajectory information. The authors propose a two-stage flow-based method: first, they construct time-parameterized transport maps from a reference distribution to the marginal distributions at each observed time point using conditional flow matching; second, they estimate the physical velocity field by regressing on synthetic trajectories generated from these maps, thereby recovering the full dynamical system. This approach uniquely identifies non-gradient physical dynamics—including rotational and cyclic behaviors—without requiring trajectory data, overcoming limitations of classical optimal transport frameworks. The method accurately reconstructs complex collective dynamics in high-dimensional settings, demonstrating its capability to capture intricate, non-equilibrium processes.

This work establishes the first rigorous theoretical foundation for neural network–based flow matching, addressing the lack of guarantees regarding convergence, generalization, and generation quality. Focusing on over-parameterized two-layer ReLU neural networks that model conditional velocity fields, the study analyzes the Wasserstein error of samples generated by the induced flow under gradient descent optimization and provides the first convergence and generalization bounds for flow matching. Furthermore, it introduces a novel generalization theory for multi-task representation learning applicable to unbounded losses. Empirical evaluations on both synthetic data and real-world image benchmarks corroborate the theoretical predictions, demonstrating that the method achieves strong convergence properties alongside high-quality sample generation.