Worst-pair hyperedge smoothness

Integrate the penalty at the hidden-state tensor immediately before the task head or before a transformer layer whose representations should be invariant within groups. Construct hyperedges from existing metadata: all augmentations of one image, all tokens belonging to one entity or span, or the top-k retrieved neighbors of an anchor. For a batch of representations H with shape [B,d], use p=2 initially. Pseudocode: penalty=0; for e in sampled_hyperedges: X=H[e]; D=pairwise_squared_distance(X); W=pairwise_weights(e); q,argmax_pair=max(W*D); penalty += q; loss=task_loss + lambda*penalty; loss.backward(); optimizer.step(). Compute pairwise distances exactly for groups up to 64 members. For larger groups, use random pair subsampling, an anchor-to-all scan, or farthest-point search. Because the max is nondifferentiable at ties, optionally replace it during a warmup period by tau*logsumexp((W*D)/tau) and anneal tau toward zero; alternatively average gradients over pairs within a small tolerance of the maximum. The mathematical quantities are the weighted maximum, p-power, and additive hyperedge decomposition. Estimate hyperedge weights, sampling frequency, lambda, and tau empirically. First experiment: train a small ViT on CIFAR-100 with four augmentations per image forming one hyperedge. Compare no regularizer, mean pairwise squared-distance regularization, and the proposed max penalty at equal batch size and FLOPs. Measure clean and corruption accuracy, worst-case augmentation distance, validation-loss variance across seeds, and training overhead. Success means lower worst-case representation discrepancy and improved corruption accuracy without a clean-accuracy loss greater than 0.5 percentage points.