发表机构
Mathematical Institute, University of Oxford; Heidelberg Institute for Theoretical Studies（牛津大学数学研究所; 海德堡理论研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对训练网络的对称性参数归因问题，通过逐参数功能敏感性分析，揭示了参数空间中实现函数空间对称性群作用的条件，并在旋转不变分类器与哈密顿神经网络上验证了相关结论。
AI 中文摘要
当网络学习到具有已知对称性的函数时，该对称性能否通过参数化进行迁移——即参数空间中是否存在一种运动，可在函数空间中实现群作用？我们将此问题表述为实现映射Φ:θ↦f_θ的提升问题，并证明：仅当函数的对称性轨道的切空间位于dΦ_θ的像空间内时，才存在光滑的参数空间作用，其中dΦ_θ的各列是单个参数的“功能敏感性”。该条件也是逐点一阶提升的充分条件。在最小二乘问题中放宽该条件时，会得到两个局部参数方向：一个沿对称性轨道延伸，另一个向等变子空间下降，其残差用于衡量参数化无法实现的部分。在一个旋转不变分类器上，我们发现这些方向会诱导出预期的函数空间运动，但仅在局部成立：重新计算的方向会跟踪轨道并减少等变缺陷，而固定的方向在训练后会偏离两者。对于在旋转对称势上训练的哈密顿神经网络（Hamiltonian neural networks），即使该架构未显式强制执行对称性，上述结论依然成立。
英文摘要
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 $Φ:θ\mapsto f_θ$, 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 $\mathrm dΦ_θ$, 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.
