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.
This paper establishes a direct analogue of the classical Coding Theorem in the setting of symmetry groups. We consider computable bijections on the set of binary strings, called symmetries and define the symmetry prior of a string as the probability that a randomly chosen symmetry from a given group has the string as its unique fixed point. We show that for any fix-retractable symmetry group, a group admitting a computable section that selects an isolating symmetry for every string, the symmetry prior is a universal lower semi-computable semi-measure. In this case, the Geometric Coding Theorem holds. We also develop a Galois connection between subgroups of G and subsets of binary strings, characterizing closed points and maximal closed subgroups, and explore the join-semilattice of dense subgroups. Our results unify algorithmic information theory with group theory and provide a framework for studying symmetry-induced complexity measures. This paper is the first in a series on Computational Algorithmic Statistics (CAS).
Post-hoc analysis of trained neural network weights often seeks to recover geometric structure directly from the parameters. We show that, for positional-encoding-equipped neural fields, the symmetry visible from weights is not the true symmetry group itself, but an observable symmetry set determined by the trained parameters, the positional encoding (PE), and readout observable. We formulate this dependence through an exact observability hierarchy, $G_{\mathrm{obs}}^{\mathrm{exact}} \subseteq G_{\mathrm{lift}}^{\mathrm{exact}}(φ) \cap G_{\mathrm{true}}$, where $G_{\mathrm{lift}}^{\mathrm{exact}}(φ)$ is the set of input transformations that the PE can exactly lift to the feature space. The hierarchy implies that even when a target function has a geometric symmetry, that symmetry may be structurally invisible to weight-level observables if the PE does not represent the corresponding transformation. We test this prediction using MLPs trained on two-dimensional signed distance functions with multiple shape symmetry groups, positional encodings, and Gram-based observables. The results show a consistent PE-dependent pattern: DyadicAxisPE supports $D_4$-sensitive readout but structurally suppresses $D_3$ rotations, TriAxisPE yields lower $D_3$ / $D_6$ readout scores under the tested Gram observables by replacing coordinate axes with three 120-degree-separated axes, and random Fourier features mainly exhibit a $π$-rotation response under these readouts. These findings show that PE design affects not only approximation behavior but also which structures are accessible to post-hoc weight-level readouts. This provides a basis for a principled observable-dependent symmetry readout.
Neural network parameter spaces are inherently non-injective, as distinct parameter configurations can realize identical functions through functional equivalence. While this symmetry is well understood in classical fully connected and convolutional models, it becomes substantially more intricate in modern attention-based architectures. Existing analyses of multihead attention have largely focused on the vanilla formulation, overlooking positional encodings that fundamentally reshape architectural symmetries. In this work, we provide a formal study of functional equivalence in Transformers with positional encodings. Focusing on the two most widely used variants--sinusoidal and rotary positional encodings (RoPE)--we show that sinusoidal encodings preserve the equivalence structure of vanilla attention, whereas rotary encodings significantly reduce the symmetry group, thereby enhancing expressivity. This offers a principled explanation for the growing prominence of RoPE in practice. We further examine how positional encodings affect linear mode connectivity, and through an alignment algorithm, empirically demonstrate that the presence and variability of connectivity across Transformer settings crucially depend on the positional encoding.
Kusha Sareen, Mohammad Pedramfar, Sékou-Oumar Kaba +2cs.LG cs.AI
Overparameterization is central to the success of deep learning, yet the mechanisms by which it improves optimization remain incompletely understood. We analyze weight-space symmetries in neural networks and show that overparameterization introduces additional symmetries that benefit optimization in two distinct ways. First, we prove that these symmetries act as a form of diagonal preconditioning on the Hessian, enabling the existence of better-conditioned minima within each equivalence class of functionally identical solutions. Second, we show that overparameterization increases the probability mass of global minima near typical initializations, making these favorable solutions more reachable. Teacher-student network experiments validate our theoretical predictions: as width increases, the Hessian trace decreases, condition numbers improve, and convergence accelerates. Our analysis provides a unified framework for understanding overparameterization and width growth as a geometric transformation of the loss landscape.