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

Beatriz Cardoso Nascimento, Marcos M. Raimundo

Track: tiny paper (up to 4 pages)

Keywords: Grassmann Manifold, Representation Learning, Latent Geometry, Fairness Auditing, Mode Collapse

TL;DR: We propose an unsupervised geometric framework that detects systematic fairness failures by lifting residuals to the Grassmann manifold.

Abstract: 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.

Anonymization: This submission has been anonymized for double-blind review via the removal of identifying information such as names, affiliations, and identifying URLs.

Submission Number: 70