Build a kernel aggregation layer whose output is a tangent vector field on the unit sphere and whose surface divergence is identically zero by construction. For each source point, use a matrix kernel obtained by applying a surface-rotated gradient in the query variable to a scalar zonal kernel; this is a differential-form version of the paper's matrix-valued construction. The layer can replace attention or message passing when the target dynamics are incompressible, such as spherical fluid…
Train a matrix-valued neural layer under an exact or near-exact Stiefel constraint while using an l1 or row-group sparsity penalty. During early training, use manifold proximal-gradient steps to identify a stable nonzero support; once the support stops changing, switch to Newton-CG steps restricted to the smooth intersection of the Stiefel tangent space and the fixed-support subspace. This can reduce the number of optimizer iterations needed to obtain sparse, well-conditioned projections.
Add a consistency loss for a neural model processing irregularly sampled functions or sequences, comparing its latent covariance representation before and after a time-coordinate warp. The target discrepancy should be near zero for affine transformations but should grow with the normalized curvature, bandwidth, and local sparsity of a nonlinear warp.
Replace ordinary token pooling or attention aggregation in a hyperbolic representation space with the point satisfying a bounded radial equilibrium law. Each token contributes a unit tangent direction multiplied by \(\tanh\) of its hyperbolic distance from the candidate, so distant outliers cannot dominate the pooled representation while nearby, geometrically consistent tokens still determine it.
Represent prototypes or attention keys by points p_i on the unit sphere and regularize their configuration with a Green-potential log-partition objective inspired by the TPMS branch-point formulation. The objective penalizes configurations whose positive and negative Gibbs-weighted potentials are concentrated in different regions, providing a smoother alternative to pairwise repulsion or uniformity losses.
When a chosen sparse support is geometrically incompatible with exact orthogonality, temporarily optimize on a nearby off-diagonally perturbed Stiefel constraint rather than forcing a singular Newton system. Anneal the perturbation to zero after the active support has stabilized, using the paper's O(||Delta||_F) KKT guarantee to control the residual of the original orthogonality-constrained problem.
Replace explicit quotient construction by a differentiable projection that removes learned group-orbit directions from both source and target features. The paper's reduction argument shows that a closed level constraint makes the restricted form horizontal, so the network can operate on invariant coordinates while retaining a measurable residual for symmetry leakage.
Add a differentiable hypergraph layer that converts invariant edge-length features into triangle areas or higher-dimensional simplex volumes before message passing. Select or weight simplices according to the singular values of the length-to-volume Jacobian, so the network receives geometrically independent features rather than many redundant or nearly degenerate measurements.
Use a learned asymmetric Finsler-like cost instead of the symmetric Euclidean distance in attention logits. The metric has a Riemannian quadratic part and a directional drift term, while a differentiable barrier enforces the strong-convexity condition derived for the paper's extended $(\alpha,\beta)$-metrics. This lets each attention head prefer one direction in feature space without producing pathological, non-convex distance landscapes.
Treat learned features on a mesh as differential forms and pool them against oriented chains using wedge or cap products instead of ordinary coordinate averaging. Couple forward and boundary features with the signed chain differential so that pooling commutes with differentiation, preserving local conservation and orientation information.
Add a centro-affine Dirichlet penalty to a neural module whose inputs or outputs lie on a sphere, such as normalized embeddings or attention directions. The penalty measures intrinsic variation under an unconditional convex-body metric while projecting out the constant and coordinate-affine modes excluded by the theorem.