Global parameter-space lifting of function-space symmetries

Determine whether, for a neural-network realisation map Phi:ThetatoB and a Lie-group action Pi on function space, every transformed function Pi(g)f_theta can be represented by some parameter vector, and whether there exists a smooth parameter-space action beta:GtimesThetatoTheta satisfying Phi(beta_gtheta)=Pi(g)Phi(theta).

Background

The paper studies whether a symmetry acting on a neural-network function can be transferred, or lifted, to the network parameters. A Lie group G acts on the input and output spaces and thereby induces an action Pi on a function space B, while the neural network is represented through a realisation map Phi:ThetatoB. A global lift would be a smooth action beta on the parameter space such that applying beta to parameters produces exactly the function obtained by applying Pi in function space.

The paper does not establish the existence of such a global action. Instead, it derives a necessary tangent-space condition and shows that the condition is sufficient only for pointwise first-order lifting. Thus, the broader problem of global or smooth local realisation of arbitrary function-space symmetries through parameter transformations remains unresolved in the general setting considered.

References

While g\in G acts naturally on f_\theta in function space, it is not clear whether \Pi(g)f_\theta is again realised by some parameter vector. Ideally, there exists a smooth action \beta:G\times\Theta\to\Theta such that \Phi(\beta_g\theta)=\Pi(g)\Phi(\theta).

— Parameter-Level Attribution of Symmetry in Trained Networks Though Parameter-Wise Functional Sensitivity  (2608.24700 - Muriithi et al., 25 Aug 2026) in Section 2, especially Section 2.1, “Symmetry Orbits and Infinitesimal Lifting”