提出统一评估扩散与流匹配采样器的两个核心性质，提升小噪声下采样精度。
Asymptotic Preservation and Uniform Accuracy of Diffusion and Flow-Matching Samplers
- 定义渐近保持与均匀精度，衡量采样器在极小噪声下的稳定性与误差
- 证明在光滑流形上，特定构造可实现三阶以上均匀精度，且误差不依赖噪声底限
- 适用于高精度生成模型调试，尤其对小噪声采样场景有指导意义
扩散与高斯插值流匹配采样器在接近终端噪声底限ε时面临奇异极限，尤其对流形支撑或低秩数据。本文研究完整采样器规范（更新规则、时间网格、终止规则）的两个性质：渐近保持（AP）指在噪声趋近零时，步数有界且离散化稳定；均匀精度（UA）阶数p表示在数值分辨率h下，终点W2误差为O(h^p)，常数与ε无关。发现对数噪声步长导致步数发散，破坏AP。若在正切换尺度a处停止稳定基求解器，并附加拟合解析正态模式的映射，则可恢复AP。在光滑紧致无边流形上，标准映射输入误差为O(a²−ε²)，零底限误差为Θ(a²)。若基求解器在有效区间具阶p的ε-一致估计，则当a=O(h^{p/2})时仍保持该阶，前提是终端转移因子有界。沿精确轨迹，后验均值恒等式D(x(σ),σ)=x(σ)−σx'(σ)可消除线性终端缺陷，支持更高阶拟合映射。三点赫米特构造在0≤ε≤a下实现均匀三阶，七点构造在ε=0时达四阶。我们据此分类典型扩散与流匹配方案的AP与UA表现。在EDM和修正流检查点上，通过分解基积分与终段补全误差，可分离并预测同种子终点误差。
原文摘要 · Abstract (English)
Diffusion and Gaussian-interpolant flow-matching samplers approach data through a terminal noise floor $\varepsilon$, a singular limit for manifold-supported or rank-deficient data. We study two properties of a complete sampler specification, comprising its update rule, time grid, and terminal rule. Asymptotic preservation (AP) means a stable and consistent zero-noise discretization with a step count bounded independently of $\varepsilon$. Uniform accuracy (UA) of order $p$ means that, at numerical resolution $h$, the endpoint $W_2$ error is $O(h^p)$ with a floor-independent constant. Bounded log-noise stepping fails AP because its step count diverges. Stopping a stable base solver at a positive switching scale $a$ and appending one map fitted to the analytic normal mode restores AP. On smooth compact boundaryless manifolds, the standard map has exact-input error $O(a^2-\varepsilon^2)$ and sharp zero-floor error $Θ(a^2)$. A base solver with a floor-uniform order-$p$ estimate on the resolved interval retains that order when $a=O(h^{p/2})$, provided the terminal transfer factor remains bounded. Along exact trajectories, the posterior-mean identity $D(x(σ),σ)=x(σ)-σx'(σ)$ cancels the linear terminal defect and enables higher-order fitted maps. A three-evaluation Hermite construction is uniformly third order for exact switching-scale input over $0\le\varepsilon\le a$, and a seven-evaluation construction is fourth order at zero. We classify representative diffusion and flow-matching specifications by AP and UA. On EDM and Rectified Flow checkpoints, a paired decomposition separates base-integration from terminal-completion error and predicts held-out same-seed endpoint errors.
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