Every idea extracted from recent arXiv mathematics papers — verified and unverified. Click an idea to open its full card; badges show the empirical verdict.
Build a graph neural network on the dual graph of a triangulated surface whose messages are transported by \(\mathfrak{S}_3\) permutation matrices associated with adjacent-face color transports. This removes dependence on arbitrary local color-label choices and gives the network an explicit representation of noncontractible topology through holonomy around cycles.
Add a spectral regularizer to a learned graph or sparse attention adjacency that penalizes violation of the paper's energy floor. The regularizer discourages adjacency matrices that retain many edges but collapse into a low-dimensional spectral structure, which may reduce graph-message-passing diversity and worsen oversmoothing.
Treat a spatial feature map or lattice-indexed embedding as a function on a d-dimensional discrete grid and penalize excessive concentration near a chosen anchor using the inverse-radial Hardy weight. Calibrate the penalty with the theorem's high-dimensional scaling 2^ell d^ell instead of selecting an arbitrary spatial L2 coefficient.
Convert an attention or MoE routing affinity matrix into a soft graph and constrain its K_r-density relative to its observed K_s-density. The regularizer penalizes pathological affinity patterns in which moderate s-way coactivation is accompanied by an implausibly low or unstable r-way coactivation.
Represent the active experts or channels of a sparse layer by a presence set and impose a reaction-style dependency graph on possible support changes. During a growth phase, activate only the least support set closed under enabled dependencies; during later pruning, allow trajectory-dependent removals but never add structurally unreachable experts. This should reduce routing churn and dead experts while preserving adaptive sparsity.