level: research
machine learning models often ignore symmetries in their input data. this paper introduces a framework to quantify what information is lost when a model fails to detect certain group actions. given a lie group acting on the input space, the authors define the null fiber: the set of group elements whose action on a point leaves the model's output unchanged. when this set is the same for all inputs, it matches the stabilizer subgroup of the function, the largest subgroup under which the model is invariant.
for smooth functions mapping to real numbers, the preimage theorem shows that null fibers have dimension at least the group dimension minus one at typical inputs, no matter the model architecture. for compact groups acting on themselves, the peter-weyl theorem gives a spectral description using fourier coefficient matrices. this connects the discarded information to the model's internal representations and provides a way to compute the stabilizer from the learned function.
the framework has practical uses in privacy and model fingerprinting. by analyzing what symmetries a model ignores, one can infer properties of the training data or identify the model. the discarded geometry can also be used to design attacks or defenses. the work bridges group theory and machine learning, offering tools to audit models for unintended invariances and to understand how data structure is lost during learning.
why it matters: it gives a mathematical tool to audit models for hidden invariances, which can leak private information or help identify model provenance.