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

Eduarda de Souza Marques; Arthur Sobrinho Ferreira da Rocha; Joao Paixao; Heudson Mirandola; Daniel Sadoc Menasche

cs.LGMar 10, 2026

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

Authors: Eduarda de Souza Marques, Arthur Sobrinho Ferreira da Rocha, Joao Paixao, Heudson Mirandola, Daniel Sadoc Menasche

Organizations: Institute of Computing, UFRJ · Institute of Mathematics, UFRJ

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.

Figures & tables

Panel of Figure 1 — Figure 1: Representative MNIST digit samples for the GSVD pipeline. (a,b) raw digits “1” and “5”; (c,d) corresponding vectors after mean-centering.

Panel of Figure 2 — Figure 2: Empirical distributions of the angle $\theta(z)$ on the MNIST test set for four digit pairs. Values closer to $0$ indicate stronger alignment with the first digit in the pair, while values closer to $90^{\circ}$ indicate stronger alignment with the second.

Panel of Figure 3 — Figure 3: Representative $H$ component directions reconstructed in the image space as 28x28 images, using the viridis colormap, obtained by the GSVD-based optimization in Section 4.3 : (a) representation of a solution from the minimization problem indicating a more “4-like” direction; (c) representation of the solution from the maximization problem indicating a more “9-like” direction; (b) reconstruction of a shared direction that encapsulates the structure of both “4” and “9”.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$d$	Ambient dimension (e.g., $784$ for $28\times 28$ MNIST images)
$p,q$	Number of columns (samples or basis vectors) used to form $A$ and $B$
$A\in\mathbb{R}^{d\times p}$ , $B\in\mathbb{R}^{d\times q}$	Dataset matrices (columns are observations in $\mathbb{R}^{d}$ )
$a_{i}$ , $b_{j}$	Columns of $A$ and $B$ (individual observations)
$\operatorname{col}(A)$ , $\operatorname{col}(B)$	Column spaces (subspaces spanned by columns of $A$ and $B$ )
$x\in\mathbb{R}^{p}$ , $y\in\mathbb{R}^{q}$	Coefficients such that $Ax$ and $By$ represent an ambient vector

Appendix

Table 1: Notation used throughout the paper.

Panel of Figure 4 — Figure 4: Representative MNIST Fashion samples for the GSVD pipeline. (a,b) raw “ T-shirt ” and “ Sneakers ”; (c,d) corresponding vectors after mean-centering difference.

Panel of Figure 5 — Figure 5: Empirical distributions of the angle $\theta$ on the test set for two Fashion-MNIST class pairs under the same GSVD pipeline used for MNIST digits (same preprocessing and the same rank choices $(p,q)$ ). (a) T-shirt/top (class 0) vs. Sneaker (class 7), where $A$ is built from T-shirt/top columns and $B$ from Sneaker columns; the separation indicates limited shared structure. (b) Sneaker (class 7) vs. Ankle boot (class 9), which shows larger overlap, consistent with more shared directions between these visually related footwear classes. As in the main text, angles near $0$ indicate stronger alignment with $A$ and angles near $90^{\circ}$ indicate stronger alignment with $B$ .

Digit pair $(A,B)$	Fisher–Rao distance (rad)
$(0,7)$	2.773465
$(1,5)$	2.731825
$(3,9)$	2.665724
$(4,9)$	2.336575

Appendix

Table 2: Fisher–Rao distance (in radians) between the class-conditional

\theta

histograms for selected MNIST digit pairs (30 bins over

[0^{\circ},90^{\circ}]

Panel of Figure 6 — Figure 6: Posterior distributions across discretized angle bins for four digit pairs in the MNIST test set. Angles near $0$ reflect stronger alignment with the first digit in the pair, whereas angles near $90$ reflect stronger alignment with the second. Translucent bins indicate angle ranges for which no samples were observed.

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We propose Rashomon Alignment (RA), a new measure to assess functional similarity between two models. Existing functional similarity measures are distributional, quantifying differences between outputs of models applied to real-world data. However, these measures can be regarded as ecologically valid only for regions in the input space represented by the available data. We introduce a geometrical perspective on functional model similarity, which estimates it across the entire data space, offering a comprehensive view of decision boundary alignment independent of any specific data distribution. We also propose geometric Rashomon Alignment as a measure of geometrical similarity, which is computed using data uniformly sampled from the instance space. We perform an experimental analysis on more than 90 datasets, examining critical cases where model alignment diverges from predictive accuracy. Our results show that geometrical and distributional alignment provide different and complementary perspectives on the similarity between models and algorithms. RA can be used for multiple purposes, including model selection, ensemble construction, and enhanced interpretability of machine learning models and algorithms.

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