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

arXiv: https://arxiv.org/abs/2608.24700 · HTML: https://mlsanity.com/paper/2608.24700/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 7 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Machine Learning Hamiltonian Dynamical Systems with Sparse and Noisy Data](https://mlsanity.com/paper/2604.17470/) (2026)

## Abstract (arXiv)

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:θmapsto fθ, 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 dPhiθ, whose columns are the 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.