What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry

arXiv:2607.13046v1 Announce Type: cross Abstract: We develop a framework for the information discarded by machine learning models whose inputs carry a Lie group action. Given a representation $\pi$ of a Lie group $G$ on a space $V$ and a learned function $f\colon V \to \mathbb{R}$, we define two objects measuring the symmetry invisible to $f$. The null fiber at a point $x \in V$ is the set $N_G(f,x) = \{g \in G : f(\pi(g^{-1}) \cdot x) = f(x)\}$ of group elements whose inverse action on $x$ is undetectable by $f$. When $N_G(f,x)$ is independent of $x$, it coincides with the stabilizer $\mathrm