[论文] Score Accuracy Along the Forward Diffusion Does Not Certify Numerical ...

论文概要

研究领域: ML
作者: Yiwei Zhou
发布时间: 2026-07-10
arXiv: 2507.08175

中文摘要

分数匹配控制前向边缘分布下的平均误差，但离散化逆时间采样器沿其自身轨迹评估学习到的分数。我们证明，小的前向边缘误差并不能保证数值稳定性。我们构造了一个单一光滑分数场，其前向边缘L²误差可任意小。学习到的逆时间过程无爆炸性，具有任意阶矩，且在路径空间全变差下可任意接近精确逆时间过程。然而其Euler-Maruyama离散化在概率下收敛，而每个正阶矩发散。因此弱收敛可以成立，即使每个Wasserstein距离W_p（p≥1）都发散。同样的失效也可能发生在单个固定的有限神经网络架构中。我们构造了一族有界、全局Lipschitz的去噪器，其前向边缘误差和路径空间全变差距离都趋于零，而它们的Euler-Maruyama端点在每个W_p下都发散。对于紧支撑数据，我们也给出一个简单的正面结果：将学习到的去噪器投影到包含支撑集的已知有界闭凸集上，可保持逐点精度，给出网格一致矩界，并在温和局部正则性下产生Wasserstein收敛。

原文摘要

Score matching controls average error under the forward marginals, but a discretized reverse-time sampler evaluates the learned score along its own trajectory. We show that small forward-marginal error does not guarantee numerical stability. We construct a single smooth score field with arbitrarily small forward-marginal L² error. The learned reverse-time process is nonexplosive, has moments of every order, and can be arbitrarily close to the exact reverse-time process in path-space total variation. Yet its Euler--Maruyama discretizations converge in probability while every positive moment diverges. Thus weak convergence can hold even though every Wasserstein distance W_p, p≥1, diverges. The same failure can occur within one fixed finite neural architecture. We construct a family of bounde...

自动采集于 2026-07-11

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