The Geometry of Memorization: Finite-Time Spectral Sensitivity as a Diagnostic for Flow Matching Models

Abstract

Continuous-time generative frameworks construct probability paths between base and target domains by optimizing time-dependent velocity fields. While theoretical targets favor straight trajectories, empirical networks develop complex path deformations. This paper presents the Finite-Time Spectral Sensitivity (FTSS) g(t), a gradient-free, forward-pass metric that exposes flow geometry by tracking the root-mean-square singular value of the state-transition matrix. Serving as a continuous proxy for stable rank, g(t) reveals a distinct geometric pathology under data scarcity: while generalizing models maintain stable effective dimensions, overfitting causes a spectral collapse. We leverage this structural phenomenon to develop an internal geometric audit based on g(t). Our framework detects generative memorization using purely internal trajectory dynamics, removing the need for external membership queries or baseline data comparison.