扩散与流匹配采样器的渐近保持后验分析

中文摘要 AI 辅助

扩散和流匹配采样器将学习到的概率流常微分方程从大噪声尺度积分到小终端下限σ_min，此时得分僵硬且流形成边界层。我们将σ_min视为奇异摄动参数，确定哪些固定步长采样器是渐近保持的，将标准作为后验审计：具有σ_min均匀系数的残差泛函，可在预训练检查点上计算，无需真实得分或精确轨迹。在终端层，σ时钟中的欧拉方法、确定性DDIM更新是唯一的层精确离散化，λ时钟仅在步长h≤h_star = 1 + W(1/e)时稳定，均匀σ^2热时钟在距数据σ_min无关的距离处停滞。在两个可解模型上，确定性采样器保持一阶均匀准确性，无log(1/σ_min)因子，对数完全归因于随机采样器的伊藤项，其路径KL与常微分方程的预算相比缩放为Λ^2/N，而常微分方程的预算为O(Λ^2/N^2)，其中Λ = log(σ_max/σ_min)。在EDM CIFAR-10检查点上，一次测量的光谱可预测跨步数、调度和噪声水平的留出残差预算，无需针对每个配置重新拟合，并在M_1 = 1.00±0.01处校准伊藤系数。时钟决定稳定性；噪声而非几何结构导致对数出现。

英文摘要

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.

发表机构

• University of Washington（华盛顿大学）

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