Every idea extracted from recent arXiv mathematics papers — verified and unverified. Click an idea to open its full card; badges show the empirical verdict.
Replace a dense token or state-mixing matrix with an inverse-capacitance operator whose couplings decay with graph distance, while introducing trainable heterogeneous diagonal capacitances to break spatial symmetries. The layer is cheap because the capacitance matrix is sparse and banded, but its inverse produces global responses with controllable locality.
Add a learned Riesz-transform branch that extracts normalized spatial gradients after diffusion by a positive parabolic operator. The diffusion branch carries smooth semantic content, while the Riesz branch represents boundaries, motion changes, and graph discontinuities. Resolvent smoothing makes the derivative branch less sensitive to feature noise than directly applying a finite difference.
Add a fixed or weakly parameterized residual mixer whose interaction between sequence positions at distance \(r\) is proportional to \(1/(r\log^2 r)\). Instead of truncating the kernel at a short radius, represent its heavy tail with dyadic distance bands and compute each band using prefix sums or block pooling, giving every token access to arbitrarily distant context at roughly \(O(L\log L)\) cost.
Replace dense grid tokens or global spectral features with coefficients of compactly supported kernels centered on a nested hierarchy of spatial points. Encode an input field into coarse-to-fine coefficients, apply a neural map to those coefficients, and decode the predicted coefficients at arbitrary query locations; the contribution from each level provides an explicit multiscale output decomposition.
Replace a dense weak pairwise interaction over tokens or graph nodes by an analytically resummed Gaussian quadratic term plus explicitly evaluated corrections containing only subsets of size at most R. Choose R from the target approximation error and interaction scale instead of fixing the interaction order heuristically. This is intended for dense global attention, graph interactions, or latent-variable energy models with individually small couplings.
Replace pointwise pair interactions between mesh cells by quadrature of the interaction kernel over the full Cartesian product of the two cells. Decompose each cell pair into convex-hull pieces and apply a Duffy-like radial transformation so the coincidence singularity is confined to one quadrature coordinate, allowing fixed Gauss-Jacobi or adaptive quadrature to produce smooth, low-variance interaction features.
Use the sharp exponential tail bound to set a local clipping threshold from a desired exceedance probability. Instead of globally clipping activations at a fixed value or percentile, clip each local window at its minimum plus B log(e/delta), where delta is the tolerated fraction of clipped entries.
Replace mean or max pooling over a set of learned element embeddings with pooling based on the metric-magnitude weighting. Pairwise distances create a globally coupled correction for redundancy, so geometrically isolated or boundary elements can contribute differently from dense clusters of nearly duplicate elements.
Partition a sequence, image, or graph into regions processed by a cheap local operator and a more expressive nonlocal operator, then couple their boundary activations with a shared continuity equation and a conservative interface-flux equation. The interface correction prevents the local and global branches from creating discontinuities or duplicated information, allowing nonlocal computation to be restricted to selected regions while preserving global consistency.