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Short summary

This paper develops a mathematical framework for the information ML models discard when inputs carry Lie group symmetry. It defines null fibers and stabilizers that measure symmetry invisible to a learned function, and shows these can be computed efficiently via Newton iteration. Applications include data masking, model fingerprinting, and privacy-preserving computation, tested on molecular property prediction under SO(3) and spherical image classification under the Möbius group. The framework applies to both classical neural networks and variational quantum circuits.

• •Defines null fibers and stabilizers to measure symmetry information discarded by ML models
• •Efficient computation via Newton iteration at cost comparable to a few gradient evaluations
• •Applications to data masking, fingerprinting, and privacy tested on molecular and spherical image tasks

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