GSVD for Geometry-Grounded Dataset Comparison: An Alignment Angle Is All You Need

• URL: http://arxiv.org/abs/2603.10283v1
• Date: Tue, 10 Mar 2026 23:58:09 GMT
• Title: GSVD for Geometry-Grounded Dataset Comparison: An Alignment Angle Is All You Need
• Abstract summary: Geometry-grounded learning asks models to respect structure in the problem domain rather than treating observations as arbitrary vectors.<n>We use the generalized singular value decomposition (GSVD) as a joint coordinate system for two subspaces.<n>We derive an interpretable *angle score* $(z) in [0, /2]$ for a sample $z$, quantifying whether z is explained relatively more by $A$, more by $B$, or comparably by both.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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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