Parameter-Level Attribution of Symmetry in Trained Networks Though Parameter-Wise Functional Sensitivity

Abstract:When a network has learned a function with a known symmetry, can that symmetry be moved through the parametrisation---is there a motion in parameter space realising the group action in function space? We formulate this as a lifting problem for the realisation map $\Phi:\theta\mapsto f_\theta$, and show that a smooth parameter-space action exists only if the tangent space to the function's symmetry orbit lies within the image of $\mathrm d\Phi_\theta$, whose columns are the \emph{functional sensitivities} of individual parameters. This condition is also sufficient for pointwise first-order lifting. Relaxing it in least squares yields two local parameter directions: one following the symmetry orbit, one descending towards the equivariant subspace, with residuals measuring what the parametrisation cannot reach. On a rotationally invariant classifier we find these directions induce their predicted function-space motion, but only locally: recomputed directions track the orbit and reduce the equivariance defect, while directions held fixed depart from both after training. The same holds for Hamiltonian neural networks trained on a rotationally symmetric potential, even though the architecture does not explicitly enforce the symmetry.

Machine Learning (cs.LG)

arXiv:2608.24700 [cs.LG]

(or arXiv:2608.24700v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.24700 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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