Manifold-Adaptive Positive Quadrature Attention

(1) Integration point: apply this to one expensive aggregation in a neural operator, point-cloud network, graph message-passing layer, or vision transformer. Let the input contain M entities with values v_i and positional or graph descriptors e_i. Compute a global latent z from a cheap pooled encoder or the previous layer. Do not use it initially for standard query-key attention; use it to select and quadrature-weight entities before the expensive value projection or message computation. (2) Pseudocode: initialize S with all entities or a fixed-weight coreset; compute phi_i=MLP_phi([e_i,z]) and b=sum_i a_i phi_i; while |S|>K, tentatively remove each candidate i, solve w=argmin_{w>=0} ||A_{S\{i}}w-b||^2+lambda||w-a_{S\{i}}||^2, and permanently remove the candidate with the smallest residual increase; at inference solve the final nonnegative quadratic program and return y=sum_{i in S} w_i v_i. For differentiable training, use a straight-through top-K mask and backpropagate through the unconstrained quadratic-program solution, or use a differentiable nonnegative solver. (3) Compute b and phi from the current latent z; estimate the subset offline on calibration batches, then train the controller and basis by minimizing task loss plus gamma||A_Sw-b||^2. The paper's mathematical contribution is the positive convex redistribution and greedy pruning; the basis, latent encoder, and K are engineering choices. (4) First experiment: use a 2D advection or Darcy neural operator and a small ViT on CIFAR-10. Compare dense aggregation, fixed top-K attention, and this method at K/M in {1/4,1/16,1/64}. Measure validation error, aggregation FLOPs, wall-clock latency, and moment residual. Success is at least 4x lower aggregation cost with less than 1 percentage point accuracy loss, plus stable nonnegative weights and lower error than random or fixed-weight selection at the same K.