Towards a Geometric Theory of Fairness: Detecting Mode Collapse on the Grassmannian Manifold

Generative models frequently exhibit mode collapse, disproportionately failing on minority subpopulations. This phenomenon is central to fair representation learning. However, detecting these failures without ground-truth labels remains an open challenge, as standard Euclidean metrics are often dominated by high-dimensional stochastic noise. In this work, we propose a geometric perspective on this problem: we hypothesize that systematic "unfairness" manifests not as magnitude errors, but as stable, low-rank subspaces in the residual field, distinct from random noise. We introduce a diagnostic framework that lifts residuals to the Grassmann manifold, allowing us to analyze the "shape" of model failures. Providing proof-of-concept evidence on MNIST, we demonstrate that our Grassmannian metric successfully isolates the structural failure modes. These preliminary results suggest that geometry-grounded tools are promising for the next generation of blind fairness auditing.