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* \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 *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.
