What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry
This paper presents a framework to analyze information discarded by machine learning models when inputs exhibit Lie group symmetries. It defines null fibers and stabilizers to measure the symmetry invisible to a model, and uses the Peter-Weyl theorem for a spectral characterization on compact groups. Efficient computation via Newton iteration is demonstrated. Applications to data masking, model fingerprinting, and privacy-preserving computation are experimentally validated on molecular property prediction under SO(3) and spherical image classification under the Möbius group PSL(2, C). The framework applies uniformly to classical neural networks and variational quantum circuits.
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[Submitted on 17 Jun 2026]
Title:What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry
View a PDF of the paper titled What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry, by Zachary P. Bradshaw
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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 $\mathrm{Stab}_G(f)$, 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.
Comments: 22 pages, 10 figures
Subjects:
Machine Learning (cs.LG); Cryptography and Security (cs.CR); Representation Theory (math.RT); Machine Learning (stat.ML)
MSC classes: 43A77 (Primary) 22E45, 68T07, 94A60 (Secondary)
Cite as: arXiv:2607.13046 [cs.LG]
(or arXiv:2607.13046v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.13046
arXiv-issued DOI via DataCite
Submission history
From: Zachary Bradshaw [view email] [v1] Wed, 17 Jun 2026 01:03:26 UTC (1,789 KB)
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