Beyond Local Linearity: Scale-Resolved Geometry of Learned Image Encoders

Abstract

Understanding how learned representations respond to finite input changes is important for characterizing their sensitivity, invariances, and robustness. Yet existing geometric analyses are predominantly local and describe only infinitesimal perturbations. We introduce a scale-resolved statistic that compares an encoder's measured feature displacement with its local linear prediction as the perturbation magnitude increases. Across diverse image encoders, we discover a characteristic plateau-rise-peak-decay profile, which we call the bump. The bump is absent at initialization, emerges early during standard training, and does not form under randomized labels or random-noise inputs. Its shape also varies with the training distribution and robustness objective. These results establish departures from local geometry as a signature of how encoder representations are shaped by learning.