AI's discarded geometry: New framework enables data masking & model fingerprinting

A new arXiv paper by Zachary P. Bradshaw introduces a mathematical framework for recovering and exploiting the geometric information that machine learning models discard during training. The work focuses on models whose inputs have inherent symmetries described by Lie groups—continuous transformation groups like rotations or scaling. Bradshaw defines two key objects: the null fiber, the set of group elements whose action on an input is undetectable by the model's output, and the stabilizer, the largest subgroup under which the model is fully invariant. These quantify exactly which symmetries are 'thrown away' by the learned function. The framework applies the preimage theorem to guarantee that null fibers have at least dim(G)-1 dimensions at generic inputs, regardless of architecture, and uses the Peter-Weyl theorem for spectral characterization on compact groups. Computation is efficient: null fiber elements can be found via Newton iteration on the orbit map at a cost comparable to a few gradient evaluations.

The practical implications are significant. Bradshaw demonstrates three applications: data masking (hiding sensitive attributes by ensuring certain symmetries are ignored), model fingerprinting (embedding unique signatures via controlled symmetry invariance), and privacy-preserving computation (using discarded geometry to prevent leakage of private information). Experiments on molecular property prediction under SO(3) rotational symmetry and spherical image classification under the Möbius group PSL(2,C) show the method works across classical neural networks and variational quantum circuits. The paper bridges representation theory, cryptography, and machine learning, offering a unified way to harness the information your model deliberately forgets—and why you might want it back.