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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 __ of a Lie group _G_ on a space _V_ and a learned function _f_ : _V _ R, we define two objects measuring the symmetry invisible to _f_ . The _null fiber_ at a point _x V_ is the set _NG_ ( _f, x_ ) = _{g G_ : _f_ ( __ ( _g[]_[1] ) _ x_ ) = _f_ ( _x_ ) _}_ of group elements whose inverse action on _x_ is undetectable by _f_ . When _NG_ ( _f, x_ ) is independent of _x_ , it coincides with the _stabilizer_ Stab _G_ ( _f_ ), the largest subgroup of _G_ under which _f_ is invariant. For smooth maps to 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 PeterWeyl 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 SO(3) and spherical image classification under the Mobius group PSL(2 _,_ C). The framework applies uniformly to classical neural networks and variational quantum circuits.