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

Eduarda Marques, Arthur Sobrinho, João Paixão, Daniel Menasche, Heudson Mirandola
Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling, PMLR 326:387-397, 2026.

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: \emph{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 $A = H C U$, $B = H S V$ 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 \emph{angle score} $\theta(z)\in[0,\pi/2]$ for a sample $z$, quantifying whether $z$ is explained relatively more by $A$, more by $B$, or comparably by both. The primary role of $\theta(z)$ is as a \emph{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 $\theta(z)$ is presented as an illustrative application of the score as an interpretable diagnostic tool.

Cite this Paper

BibTeX
@InProceedings{pmlr-v326-marques26a, title = {GSVD for Geometry-Grounded Dataset Comparison: An Alignment Angle Is All You Need}, author = {Marques, Eduarda and Sobrinho, Arthur and Paix\~{a}o, Jo\~{a}o and Menasche, Daniel and Mirandola, Heudson}, booktitle = {Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling}, pages = {387--397}, year = {2026}, editor = {Pouplin, Alison and Vadgama, Sharvaree and Bekkers, Erik and Kaba, Sékou-Oumar and Lawrence, Hannah and Lecha, Manuel and Baker, Elizabeth and Suk, Julian and Walters, Robin and Tomczak, Jakub and Jegelka, Stefanie}, volume = {326}, series = {Proceedings of Machine Learning Research}, month = {26 Apr}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v326/main/assets/marques26a/marques26a.pdf}, url = {https://proceedings.mlr.press/v326/marques26a.html}, 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: \emph{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 $A = H C U$, $B = H S V$ 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 \emph{angle score} $\theta(z)\in[0,\pi/2]$ for a sample $z$, quantifying whether $z$ is explained relatively more by $A$, more by $B$, or comparably by both. The primary role of $\theta(z)$ is as a \emph{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 $\theta(z)$ is presented as an illustrative application of the score as an interpretable diagnostic tool. } }
Endnote
%0 Conference Paper %T GSVD for Geometry-Grounded Dataset Comparison: An Alignment Angle Is All You Need %A Eduarda Marques %A Arthur Sobrinho %A João Paixão %A Daniel Menasche %A Heudson Mirandola %B Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling %C Proceedings of Machine Learning Research %D 2026 %E Alison Pouplin %E Sharvaree Vadgama %E Erik Bekkers %E Sékou-Oumar Kaba %E Hannah Lawrence %E Manuel Lecha %E Elizabeth Baker %E Julian Suk %E Robin Walters %E Jakub Tomczak %E Stefanie Jegelka %F pmlr-v326-marques26a %I PMLR %P 387--397 %U https://proceedings.mlr.press/v326/marques26a.html %V 326 %X 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: \emph{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 $A = H C U$, $B = H S V$ 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 \emph{angle score} $\theta(z)\in[0,\pi/2]$ for a sample $z$, quantifying whether $z$ is explained relatively more by $A$, more by $B$, or comparably by both. The primary role of $\theta(z)$ is as a \emph{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 $\theta(z)$ is presented as an illustrative application of the score as an interpretable diagnostic tool.
APA
Marques, E., Sobrinho, A., Paixão, J., Menasche, D. & Mirandola, H.. (2026). GSVD for Geometry-Grounded Dataset Comparison: An Alignment Angle Is All You Need. Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling, in Proceedings of Machine Learning Research 326:387-397 Available from https://proceedings.mlr.press/v326/marques26a.html.

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