Critical-depth sparse attention

(1) Integrate the method immediately before sparse-window attention in a 2D vision or document transformer. Let the router output a score for each candidate dyadic rectangle at several spatial scales. Prune candidates to an antichain: if one rectangle strictly contains another, retain only the higher-scoring one or assign the contained rectangle to a separate routing group. (2) Construct the active mask Omega from visible or nonpadded tokens. Approximate its dyadic halo with max-pooling or average-pooling over rectangular kernels at all available dyadic scales. For every candidate I, repeatedly double only its width to compute e1 and repeatedly double only its height to compute e2; stop when the corresponding ancestor is outside the halo. Compute alpha_I = (e1+1)^(-s)(e2+1)^(-(1-s)). Initialize H to zeros over tokens, then execute for I in candidates: H[tokens(I)] += alpha_I. Penalize router scores using score[I] -= lambda * alpha_I * mean(H[tokens(I)]), or add the exponential loss L_pack = mean(exp(tau*s*(1-s)*H)). (3) The paper supplies the complementary exponent structure and the reason exponential overlap control is plausible. The halo threshold, lambda, tau, and maximum depth are engineering parameters; estimate e1 and e2 with integer ancestor lookups and compute H with fused scatter-add. (4) First test a small Swin-like model on CIFAR-100 or ImageNet-100. Compare fixed nonoverlapping windows, unconstrained learned multiscale windows, and antichain routing with critical-depth packing at equal selected-token FLOPs. Track validation accuracy, peak activation memory, maximum and 99th-percentile H, router entropy, attention FLOPs, and loss-curve variance. Success means substantially lower worst-case overlap and memory without accuracy loss, or faster convergence at equal sparse-attention cost.