A General Topological Symmetry Index for Critical Orbits In Neural Network Loss Landscapes

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 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.

Parameter space symmetries are important for understanding neural networks' loss landscape, training dynamics, and generalization. However, systematically identifying these symmetries remains a challenge. In this paper, we formalize data-dependent parameter symmetries and characterize loss invariance and the group-action axioms through infinitesimal conditions, which provide objectives for jointly learning group generators and nonlinear action maps. Our framework systematically uncovers parameter symmetries, including previously unknown ones. To study larger networks, we establish conditions under which subnetwork symmetries extend to the full model. The same construction gives an explicit family of finite-batch symmetries, providing both analytical examples and a foundation for discovery through small subnetworks. Using the infinitesimal characterization and subnetwork construction, we implement a framework for automated discovery of parameter symmetries, and successfully uncovered symmetries in various architectures, including pretrained transformer models.

Deep neural networks are highly over-parameterized, and different parameter values represent the same predictive function. This makes their effective complexity difficult to measure using only the number of parameters or the rank of the Hessian. Singular Learning Theory addresses this issue through the local learning coefficient (LLC), which characterizes the effective complexity of a model near a given solution. Existing methods for estimating the LLC often rely on posterior sampling, which can be computationally expensive for large neural networks. This makes accurate LLC estimation difficult at scale. In this work, we use known structures in the model to simplify the analysis and make LLC estimation more tractable. Specifically, we study the LLC of a graph attention model by exploiting symmetries in both the graph structure and the attention parameters. An analytic framework through a teacher--student setting, and explicit LLC estimates after considering the symmetry--induced degeneracies are developed.