在流形上用噪声修正得分匹配，提升对地球科学数据的建模精度。
Rao-Blackwellized Score Matching on Manifolds
- 通过最近点投影构建最优条件期望目标，修正低噪声下方差发散问题。
- 小噪声展开显示一阶恢复真实流形得分，二阶偏差来自曲率与嵌入几何项。
- 在二维球面上两项几何项恰好为零，解释了为何直接方法在地球数据上表现好。
我们研究数据来自嵌入流形 $M \subset \mathbb{R}^D$ 时的去噪得分匹配（DSM）方法。当环境噪声为高斯分布时，发现目标函数方差随噪声尺度减小而发散，因此通过回归到给定流形上最近点投影的条件期望，得到 $L^2$-最优的 Rao-Blackwellized 目标。我们计算该目标的小噪声展开，发现其在一阶上恢复真实的内蕴黎曼得分，二阶偏差包含一个 Tweedie 项以及两个依赖于流形嵌入方式的几何项：作用于内蕴得分的曲率算子，以及由嵌入的第二基本形式空间变化产生的附加漂移项。在超球面情形下，推导出简化公式，并证明在 $S^2$ 上这两项几何项精确为零，为先前工作中环境 DSM 在真实地球科学球面数据上表现媲美内蕴方法提供了理论解释。
原文摘要 · Abstract (English)
We study denoising score matching (DSM) when data are drawn from an embedded manifold $M \subset \mathbb{R}^D$. We show that under ambient Gaussian corruption, the target has variance that diverges as the noise scale decreases and correct for it by regressing against the conditional expectation given the nearest point projection on the manifold: the $L^2$-optimal Rao-Blackwellized target. We then compute the small-noise expansion of this target and show that it recovers the true intrinsic Riemannian score to first order, with a second-order bias from a Tweedie term and two geometric terms dependent on how the manifold is embedded in ambient space: a curvature operator acting on the intrinsic score, and an additive drift generated by the spatial variation of the embedding's second fundamental form. On hyperspheres, we derive a simplified formula and show that both geometric terms vanish exactly on $S^2$, offering a theoretical explanation for why ambient DSM performs comparably to intrinsic methods on real Earth science spherical data in prior work.
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