flow-matching likelihood estimation

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 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.

研究探讨了在什么情况下条件流匹配可以替代点对点负对数似然，通过分解线性高斯路径的端点NLL，揭示了CFM估计的有效性和局限性。

This study addresses the scalability and statistical validity bottlenecks in likelihood approximation and inference for complex simulation models by proposing a novel framework based on aggregated normalizing flow chains. Methodologically, it integrates information-theoretic formalization with sequential decision-making paradigms to construct flexible probability distributions through the sequential optimization of bijective transformation parameters. Furthermore, an empirical likelihood estimator under moment constraints is employed to iteratively update and aggregate the global flow parameters. This research establishes a surrogate model that simultaneously ensures computational feasibility and statistical power, enabling efficient parameter exploration, hypothesis testing, and uncertainty quantification. Ultimately, the proposed approach provides a reliable Bayesian inference solution for complex systems.

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.

Standard flow matching neglects variable dependencies, introducing bias in multivariate time series anomaly detection. To address this limitation, this work proposes GRASP, a framework that innovatively integrates graph structures with flow matching. Specifically, it constructs closed-form spectral paths via hyperbolic interpolation to optimize probability distributions, and incorporates the principle of least action alongside multi-source weighted velocity prediction for efficient anomaly detection, supported by theoretical guarantees of Laplacian basis invariance. Extensive experiments on four benchmark datasets demonstrate that GRASP significantly outperforms existing models, thoroughly validating the effectiveness of the proposed spectral path construction and weighting mechanisms.

This work addresses the challenge of efficiently and accurately estimating the divergence of the probability flow ordinary differential equation (PF-ODE) in diffusion and flow-based generative models, where existing approaches are either computationally expensive or suffer from high variance. The authors propose StAD, a novel method that, for the first time, incorporates the Langevin–Stein operator into divergence distillation, enabling accurate learning of the PF-ODE divergence without explicit Jacobian computation. They theoretically show that the learned vector field belongs to the Stein class under suitable conditions. By combining function approximation with regularization techniques, StAD significantly reduces estimation variance and accelerates likelihood evaluation on benchmarks such as CIFAR-10 and ImageNet, while demonstrating broad applicability across diverse generative modeling frameworks.

This study addresses the ill-posedness and instability of probability flow ODEs (PF-ODEs) when employed as deterministic samplers. Grounded in the Fokker-Planck equation, regularized Lagrangian flows, and score matching theory, we develop a bilateral divergence control framework to analyze the well-posedness conditions of PF-ODEs. This analysis reveals intrinsic connections between sampling errors and network architectures, while elucidating the theoretical justification for early stopping mechanisms and the mismatch between density-weighted matching and velocity errors. Ultimately, this work proposes a set of constraint-preserving and stable design principles for invertible diffusion models. The validity of these theoretical findings is confirmed through controlled numerical experiments.

This work proposes a novel approach to time series anomaly detection that addresses the limitations of traditional observation-likelihood-based methods, which often fail to capture structured temporal dynamics and misclassify anomalies as normal patterns. By introducing inductive biases into the latent space of conditional normalizing flows, the method models time series as discrete-time state-space systems, enforcing latent trajectories to conform to prescribed dynamical laws. Anomalies are then defined as deviations from these expected dynamics. The approach frames anomaly detection as a goodness-of-fit test for dynamic consistency—a formulation introduced here for the first time—and evaluates compliance of latent trajectories accordingly. Experiments on both synthetic and real-world datasets demonstrate its effectiveness in detecting anomalies in frequency, amplitude, and noise characteristics, achieving high detection performance alongside strong interpretability.

该研究通过控制Radon-Nikodym导数来生成异常值，解决了现有方法不能显式控制生成样本似然性的问题，利用扩散模型中的分数函数修改实现。