研究有界域反射扩散中边界项学习问题,发现边界误差可被硬反射掩盖。
Should the Boundary Term Be Learned in Reflected Diffusion? Conormal Trace and Reflection Masking

- 提出用共法向迹约束边界处得分,避免传统方法遗漏的边界项误差
- 在超矩形上无需额外参数即可精确表示真实得分,错误值会引入不可消除的偏差
- 揭示硬反射可能掩盖边界得分错误,导致生成质量与得分精度脱节
我们研究有界域上反射扩散的得分学习问题。反射虽能保持轨迹可行性,但无法保证学习到的得分满足前向过程隐含的边界行为。通过隐式得分匹配,分部积分会产生一个边界项,该边界项在每个边界点仅依赖于一个标量:得分的扩散加权法向分量(即共法向迹)。无通量条件固定了这一值,而其余边界分量仍自由;在各向异性扩散下,其通常不同于普通法向得分分量。在超矩形上,我们的参数化方式无需额外可训练参数或随机边界估计器即可强制满足所需迹,在正则性假设下可表示真实得分;若固定错误值,则误差无法通过更多数据消除。我们将该构造扩展至单纯形和多边形区域,并识别出反射掩蔽现象:即使学习到的迹错误,硬反射仍可保持样本可行性,导致后反射评估指标隐藏误差。实验表明,在反射频率较低、各向异性扩散及约束交集附近质量较高时区分最明显;在完全反射条件下,最终样本分布改善不一致,说明硬修复可能掩盖边界得分误差,使得分准确度与下游生成质量脱钩。
原文摘要 · Abstract (English)
We study score learning for reflected diffusion on bounded domains. Reflection keeps trajectories feasible but does not ensure that the learned score satisfies the boundary behavior implied by the forward process. With implicit score matching, integration by parts leaves a boundary term, and we show that it depends on one scalar at each boundary point: the diffusion- weighted normal component of the score, or conormal trace. The no-flux condition fixes this value while leaving the re- maining boundary components unrestricted; under anisotropic diffusion it generally differs from the ordinary normal score component. On hyperrectangles, our parametrization enforces the required trace without additional trainable parameters or a stochastic boundary estimator and, under regularity assump- tions, can represent the true score, whereas fixing an incorrect value creates an error that more data cannot remove. We ex- tend the construction to simplices and polygonal domains and identify reflection masking: hard reflection can keep samples feasible even when the learned trace is wrong, so post-reflection metrics may hide the error. Experiments show the clearest separation with less frequent reflection, anisotropic diffusion, and mass near intersections of constraints; under full reflection, final sample placement improves inconsistently, illustrating how hard repair can mask boundary-score errors and decouple score accuracy from downstream generation quality.
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