GSVD for Geometry-Grounded Dataset Comparison: An Alignment Angle Is All You Need
arXiv:2603.10283v1 Announce Type: new 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 geome
arXiv:2603.10283v1 Announce Type: new 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.
Executive Summary
This article introduces a novel approach to comparing geometry-grounded datasets using the generalized singular value decomposition (GSVD). The authors propose an alignment angle score, θ(z), which measures the relative contribution of each dataset to a sample. The score is derived from the GSVD factors and serves as a per-sample geometric diagnostic. The article presents an illustrative application of the score as a binary classifier on the MNIST dataset. The authors' innovative use of GSVD and θ(z) provides a new perspective on comparing datasets, enabling the detection of subtle differences in their structural relationships. The approach has potential applications in various fields, including computer vision, natural language processing, and data analysis.
Key Points
- ▸ The article introduces the GSVD as a joint coordinate system for comparing two datasets.
- ▸ The authors propose an alignment angle score, θ(z), which measures the relative contribution of each dataset to a sample.
- ▸ The score is derived from the GSVD factors and serves as a per-sample geometric diagnostic.
- ▸ The article presents an illustrative application of the score as a binary classifier on the MNIST dataset.
- ▸ The approach has potential applications in various fields, including computer vision, natural language processing, and data analysis.
Merits
Innovative Application of GSVD
The authors' use of GSVD as a joint coordinate system for comparing datasets is innovative and provides a new perspective on the problem.
Interpretable Diagnostic Tool
The alignment angle score, θ(z), serves as a per-sample geometric diagnostic, enabling the detection of subtle differences in the structural relationships between datasets.
Potential Applications
The approach has potential applications in various fields, including computer vision, natural language processing, and data analysis.
Demerits
Limited Scope
The article focuses on a specific problem and may not be directly applicable to other areas of dataset comparison.
Computational Complexity
The GSVD computation may be computationally expensive, which could limit the approach's scalability for large datasets.
Expert Commentary
The article presents a novel and innovative approach to comparing geometry-grounded datasets using the GSVD. The authors' use of the alignment angle score, θ(z), as a per-sample geometric diagnostic provides a new perspective on the problem. However, the approach may have limitations, such as limited scope and computational complexity. Nevertheless, the potential applications of the approach make it an exciting contribution to the field. As an expert, I recommend further research into the scalability and applicability of the GSVD-based approach to larger and more complex datasets.
Recommendations
- ✓ Investigate the scalability of the GSVD-based approach to larger and more complex datasets.
- ✓ Explore the extension of the approach to multimodal learning and transfer learning.