conditional flow matching

This work investigates the theoretical relationship between Rectified Flows (RF) and Optimal Transport (OT), clarifying conditions under which they are equivalent—and exposing key limitations of prevailing claims. Method: Through rigorous theoretical analysis and constructive counterexamples, we examine the asymptotic convergence of RF to OT maps under gradient constraints, characterize the intrinsic velocity-field invariance of RF, and derive closed-form solutions for Gaussian and Gaussian mixture distributions. Contribution/Results: We prove that the widely cited result—“RF converges asymptotically to the OT map under gradient constraints”—holds only under stronger assumptions than previously stated (e.g., existence of jointly convex potential functions); enforcing such constraints generally fails to recover the true OT map. We establish the first systematic characterization of RF’s velocity-field invariance and provide explicit analytical solutions for canonical distributions. Finally, we derive necessary and sufficient conditions for RF–OT equivalence, delivering critical theoretical guidance—and precise boundary conditions—for integrating RF-based generative modeling with OT theory.

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 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 paper addresses the challenges in Bayesian posterior inference—namely, reliance on likelihood evaluation, computational inefficiency, and difficulty modeling complex posterior structures. We propose a generative multivariate posterior sampling method based on flow matching. Our approach learns a dynamic block-triangular velocity field in the joint data-parameter space to construct a deterministic Brenier transport map, enabling likelihood-free and efficient posterior sampling. By imposing monotonicity constraints, the map is guaranteed to align with Monge–Kantorovich data-depth level sets, thereby yielding geometrically interpretable Bayesian credible sets endowed with frequentist guarantees—specifically, posterior consistency and credible set convergence. Leveraging conditional flow matching and invertible time integration to solve for the vector field, our method significantly outperforms GANs and diffusion models in computational efficiency while accurately capturing high-dimensional, non-Gaussian, and multimodal posterior geometries.

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 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.

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 demonstrates that diffusion models, score-based generative models, and flow matching methods—despite their apparent formal differences—share a unified continuous-time generative mechanism. By constructing a measure-theoretic framework, the paper unifies these approaches as learning time-dependent vector fields that transport a reference distribution to the data distribution, with distributional evolution governed by the continuity equation and the Fokker–Planck equation. It establishes, for the first time under a common perspective, the equivalence and distinctions among the three paradigms, clarifies the relationship between probability flow ODEs and stochastic backward dynamics, and identifies flow matching as essentially a velocity field regression problem. The study further provides a systematic comparison of objective functions, sampling strategies, and discretization errors, links the framework to Schrödinger bridges and entropy-regularized optimal transport, and summarizes theoretical guarantees and open challenges regarding approximation capacity, stability, and scalability.

This work addresses the challenge of generative modeling on measure spaces, particularly the transport of measures over measures (metameasures), by introducing the Wasserstein-on-Wasserstein (WoW) framework. It extends flow matching to the Wasserstein space of probability measures for the first time, leveraging a nested optimal transport plan—comprising both outer and inner couplings—to naturally induce a velocity field and define a deterministic dynamical system. By integrating nested Wasserstein geometry, sliced Wasserstein distances, and linearized approximations, the method achieves substantially improved computational efficiency and numerical stability. The resulting generative trajectories are nearly straight in the Wasserstein sense, yielding state-of-the-art performance with strong theoretical coherence in point cloud and set generation tasks.