GSVD for Geometry-Grounded Dataset Comparison: An Alignment Angle Is All You Need
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908878050951168 |
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| author | Marques, Eduarda de Souza da Rocha, Arthur Sobrinho Ferreira Paixao, Joao Mirandola, Heudson Menasche, Daniel Sadoc |
| author_facet | Marques, Eduarda de Souza da Rocha, Arthur Sobrinho Ferreira Paixao, Joao Mirandola, Heudson Menasche, Daniel Sadoc |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_10283 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | GSVD for Geometry-Grounded Dataset Comparison: An Alignment Angle Is All You Need Marques, Eduarda de Souza da Rocha, Arthur Sobrinho Ferreira Paixao, Joao Mirandola, Heudson Menasche, Daniel Sadoc Machine Learning 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. |
| title | GSVD for Geometry-Grounded Dataset Comparison: An Alignment Angle Is All You Need |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2603.10283 |