Completely-Bounded Schur Mask

(1) Integration point: insert the regularizer into an attention or graph-neural-network layer whose learned interaction mask m_theta is applied entrywise to an n by n logits, adjacency, or message matrix. Use n=4 and p=4 initially, matching the paper's explicit construction; for larger n, use repeated 4 by 4 blocks or a low-rank learned mask. (2) Pseudocode: sample K random complex matrices X of shape (s*n,s*n), partition each into n by n blocks X_jk of shape s by s, compute Y_jk=m_theta[j,k]*X_jk, and estimate r_s=max_k ||Y||_{S_4}/(||X||_{S_4}+epsilon). Separately estimate r_1 using scalar matrices. Add lambda*max(0,r_s-r_1) to the task loss for s in {2,4,8}. Normalize m_theta after each update so r_1 remains below a chosen scale c. Complex values can be implemented as paired real and imaginary channels. (3) Estimate the norms with automatic differentiation: initialize X randomly, normalize X by its Schatten norm, perform several ascent steps on log(||M_m^(s)(X)||_S4+epsilon), then retain the largest ratio over random restarts. The paper supplies the exact Schur and amplification operations; the finite-s norm and maximizer are estimated numerically. (4) First experiment: train a four-token toy transformer or four-node graph attention network on a synthetic relational classification task. Compare an unconstrained mask, scalar operator-norm clipping, and cb-regularization. Evaluate with channel widths s=1,2,4,8 and duplicated or correlated feature channels. Success is a lower worst-case ratio r_4/r_1, fewer activation explosions or NaNs, and equal or better accuracy. A useful target is at least a 20 percent reduction in amplified output norm at equal training FLOPs and validation accuracy.