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

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 $f:V\to \mathbb{R}$, 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 ${\mathrm{Stab}}_G(f)$, the largest subgroup of $G$ under which $f$ is invariant. For smooth maps to $\mathbb{R}$, the preimage theorem guarantees that null fibers have dimension at least $\dim G-1$ at generic inputs, regardless of architecture. For compact groups acting on themselves, the Peter--Weyl theorem yields a spectral characterization of both objects in terms of the Fourier coefficient matrices of $f$. We show that null fiber elements can be computed efficiently via Newton iteration on the orbit map, at a cost comparable to a few gradient evaluations. Applications to data masking, model fingerprinting, and privacy-preserving computation are developed and tested experimentally on molecular property prediction under $\mathrm{SO}(3)$ and spherical image classification under the Möbius group $\mathrm{PSL}(2,\mathbb{C})$. The framework applies uniformly to classical neural networks and variational quantum circuits.

22 pages, 10 figures