标题： 使用Koopman算子展开生成流：快速且可解释的采样 显示英文标题

标题： Unfolding Generative Flows with Koopman Operators: Fast and Interpretable Sampling

摘要： 条件流匹配（CFM）为训练连续时间生成模型提供了一个无需模拟的框架，连接了扩散和流基于的方法。 然而，从CFM采样仍然依赖于数值求解非线性ODE，这可能计算成本高且难以解释。 最近的替代方法通过轨迹直线化、小批量耦合或蒸馏来提高采样速度。 然而，这些方法通常无法揭示生成过程的底层\textit{结构}。 在本工作中，我们提出加速CFM并通过整合Koopman算子理论引入其动态的可解释表示，该理论在可观测值的学习空间中将非线性流建模为线性演化。 我们引入了一种无解码器的Koopman-CFM架构，该架构学习一个使生成动态变为线性的嵌入，通过矩阵指数实现闭式、一步采样。 这在受控的2D数据集和真实世界基准（MNIST，Fashion-MNIST（F-MNIST），多伦多人脸数据集（TFD））上展示了传统CFM的显著加速。 与之前的方法不同，我们的方法导致了一个结构良好的Koopman生成器，其谱特性、特征值和特征函数提供了分析生成行为（如时间缩放、模式稳定性以及Koopman潜在空间中的分解）的原则性工具。 通过结合采样效率与分析结构，增强Koopman的流匹配为快速且可解释的生成建模提供了一个潜在的步骤。 显示英文摘要

摘要： Conditional Flow Matching (CFM) offers a simulation-free framework for training continuous-time generative models, bridging diffusion and flow-based approaches. However, sampling from CFM still relies on numerically solving non-linear ODEs which can be computationally expensive and difficult to interpret. Recent alternatives address sampling speed via trajectory straightening, mini-batch coupling or distillation. However, these methods typically do not shed light on the underlying \textit{structure} of the generative process. In this work, we propose to accelerate CFM and introduce an interpretable representation of its dynamics by integrating Koopman operator theory, which models non-linear flows as linear evolution in a learned space of observables. We introduce a decoder-free Koopman-CFM architecture that learns an embedding where the generative dynamics become linear, enabling closed-form, one-step sampling via matrix exponentiation. This results in significant speedups over traditional CFM as demonstrated on controlled 2D datasets and real-world benchmarks, MNIST, Fashion-MNIST (F-MNIST), and the Toronto Face Dataset (TFD). Unlike previous methods, our approach leads to a well-structured Koopman generator, whose spectral properties, eigenvalues, and eigenfunctions offer principled tools for analyzing generative behavior such as temporal scaling, mode stability, and decomposition in Koopman latent space. By combining sampling efficiency with analytical structure, Koopman-enhanced flow matching offers a potential step toward fast and interpretable generative modeling.