用几何对齐角量化数据集间相似性，直观判断样本归属来源。
GSVD for Geometry-Grounded Dataset Comparison: An Alignment Angle Is All You Need
- 基于广义奇异值分解构建共享坐标系，分离共有的与特定方向。
- 提出角度得分θ(z)，在[0, π/2]区间内衡量样本更依赖哪个数据集。
- 可作为每样本的几何诊断工具，适用于模型偏差分析与数据对比。
几何接地学习要求模型尊重问题域中的结构，而非将观测视为任意向量。受此启发，我们重新审视一种经典但未被充分利用的数据集比较方法：两个数据矩阵间的线性关系，通过共享空间中的共张约束 $Ax = By = z$ 表达。为实现该比较，我们采用广义奇异值分解（GSVD）作为两个子空间的联合坐标系。具体地，利用 GSVD 形式 $A = HCU$, $B = HSV$，其中 $C^{ op}C + S^{ op}S = I$，通过 $(C, S)$ 的对角结构分离共享与数据集特有方向。由此导出一个可解释的*角度得分* $θ(z) \in [0, π/2]$，用于量化样本 $z$ 相对更由 $A$、更由 $B$ 还是两者共同解释。$θ(z)$ 主要作为*每样本的几何诊断*工具。我们在 MNIST 上通过角度分布和代表性 GSVD 方向展示该得分行为。此外，基于 $θ(z)$ 构建的二分类器展示了其作为可解释诊断工具的示例应用。
原文摘要 · Abstract (English)
Geometry-grounded learning asks models to respect structure in the problem domain rather than treating observations as arbitrary vectors. Motivated by this view, we revisit a classical but underused primitive for comparing datasets: linear relations between two data matrices, expressed via the co-span constraint $Ax = By = z$ in a shared ambient space. To operationalize this comparison, we use the generalized singular value decomposition (GSVD) as a joint coordinate system for two subspaces. In particular, we exploit the GSVD form $A = HCU$, $B = HSV$ with $C^{\top}C + S^{\top}S = I$, which separates shared versus dataset-specific directions through the diagonal structure of $(C, S)$. From these factors we derive an interpretable *angle score* $θ(z) \in [0, π/2]$ for a sample $z$, quantifying whether z is explained relatively more by $A$, more by $B$, or comparably by both. The primary role of $θ(z)$ is as a *per-sample geometric diagnostic*. We illustrate the behavior of the score on MNIST through angle distributions and representative GSVD directions. A binary classifier derived from $θ(z)$ is presented as an illustrative application of the score as an interpretable diagnostic tool.
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