提出一种欧拉视角下的直线采样方法，让生成过程更易数值积分。
An Eulerian Perspective on Straight-Line Sampling
- 从速度条件出发，构建直线路程的偏微分方程判据。
- 证明仅当终点确定耦合时，才能实现完全直线流动。
- 为生成模型设计更易积分的流形提供通用指导。
我们研究生成建模中的动态测度传输：具体而言，是通过随机过程诱导的流，连接指定的源分布与目标分布。该过程的速度条件期望定义了一个常微分方程（ODE），其流映射可实现所需的测度传输。我们探讨了哪些过程能产生直线流——即点对点加速度为零、可用一阶方法精确积分的流。我们给出了直线性的一个简洁偏微分方程特征：条件加速度与加权协方差（雷诺）张量散度之间的平衡。基于此视角，我们完整刻画了时间仿射插值，并证明直线性恰好发生在确定性端点耦合条件下。同时，我们推导出一般过程中流几何的必要约束，为设计更易积分的传输路径提供广泛指导。
原文摘要 · Abstract (English)
We study dynamic measure transport for generative modeling: specifically, flows induced by stochastic processes that bridge a specified source and target distribution. The conditional expectation of the process' velocity defines an ODE whose flow map achieves the desired transport. We ask \emph{which processes produce straight-line flows} -- i.e., flows whose pointwise acceleration vanishes and thus are exactly integrable with a first-order method? We provide a concise PDE characterization of straightness as a balance between conditional acceleration and the divergence of a weighted covariance (Reynolds) tensor. Using this lens, we fully characterize affine-in-time interpolants and show that straightness occurs exactly under deterministic endpoint couplings. We also derive necessary conditions that constrain flow geometry for general processes, offering broad guidance for designing transports that are easier to integrate.
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