Score-based diffusion models are typically trained by minimizing the $L^2$ score matching error, and standard theoretical analyses rely on this quantity to bound the sampling discrepancy between the learned and target distributions. We show the $L^2$ score error is not the right intrinsic measure of marginal distributional quality: a learned diffusion model can incur arbitrarily large $L^2$ score error while perfectly matching the target distribution. By decomposing score errors into a gradient and a solenoidal component (a Helmholtz-Hodge decomposition), we identify the geometric reason behind this: only the gradient component enters the marginal Fokker-Planck dynamics, while the solenoidal component is structurally invisible. We make this precise in three results. First, building on the corrected geometry, we prove an impossibility result: no monotone function of the $L^2$ score error can uniformly lower bound any divergence between the learned and target distributions. Second, we derive an upper bound on the Kullback-Leibler divergence that depends only on the observable gradient component of the error, tightening the standard Girsanov bound and identifying its looseness as the cost of operating on path-space rather than marginal-space dynamics. Third, we give a tractable estimator of the gradient component via a dual Sobolev identity, which is shown to empirically correlate substantially better with sample quality than the full $L^2$ error.

翻译：基于分数的扩散模型通常通过最小化$L^2$分数匹配误差进行训练，标准理论分析依赖该量来约束学习分布与目标分布之间的采样差异。我们证明$L^2$分数误差并非衡量边际分布质量的恰当内在指标：学习到的扩散模型在完美匹配目标分布时，可能产生任意大的$L^2$分数误差。通过将分数误差分解为梯度分量与螺线管分量（亥姆霍兹-赫奇分解），我们揭示了其背后的几何原因：仅梯度分量影响边际福克-普朗克动力学，而螺线管分量在结构上不可见。我们通过三个结果精确阐明这一现象。首先，基于修正后的几何结构，我们证明了一个不可能性结果：$L^2$分数误差的任何单调函数均无法一致地给出学习分布与目标分布之间任意散度的下界。其次，我们推导了仅依赖于可观测梯度误差分量的库尔贝-莱布勒散度上界，该界收紧了标准吉尔萨诺夫界，并指出其松弛性源于在路径空间而非边际空间动力学上操作的成本。第三，我们通过对偶索博列夫恒等式给出了梯度分量的可计算估计量，实验表明该估计量与样本质量的相关性显著优于完整的$L^2$误差。