Gauge-invariant representation holonomy

Vasileios Sevetlidis, George Pavlidis

Keywords: Representation learning, Gauge invariance, Holonomy, Geometric deep learning, Robustness

TL;DR: We introduce representation holonomy, a gauge-invariant statistic that captures the path dependence of learned features, revealing geometric structure in deep networks linked to robustness beyond pointwise similarity measures.

Abstract: Deep networks learn internal representations whose geometry—how features bend, rotate, and evolve—affects both generalization and robustness. Existing similarity measures such as CKA or SVCCA capture pointwise overlap between activation sets, but miss how representations change along input paths. Two models may appear nearly identical under these metrics yet respond very differently to perturbations or adversarial stress. We introduce representation holonomy, a gauge-invariant statistic that measures this path dependence. Conceptually, holonomy quantifies the “twist” accumulated when features are parallel-transported around a small loop in input space: flat representations yield zero holonomy, while nonzero values reveal hidden curvature. Our estimator fixes gauge through global whitening, aligns neighborhoods using shared subspaces and rotation-only Procrustes, and embeds the result back to the full feature space. We prove invariance to orthogonal (and affine, post-whitening) transformations, establish a linear null for affine layers, and show that holonomy vanishes at small radii. Empirically, holonomy scales with loop radius and depth, separates models that appear similar under CKA, and correlates with adversarial and corruption robustness across training regimes. It also tracks training dynamics as features form and stabilize. Together, these results position representation holonomy as a practical and scalable diagnostic for probing the geometric structure of learned representations beyond pointwise similarity.

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Submission Number: 3762