Hive-Rhombus Concavity Regularizer

Use the regularizer on any scalar tensor with a meaningful two-dimensional index, most directly a ViT attention-logit map \(H\in\mathbb R^{B\times heads\times N\times N}\) after reshaping tokens into an ordered patch lattice, or on a cross-attention score surface between two ordered sequences. Do not apply it to arbitrary channel coordinates. Build a triangular mask for valid \((i,j)\) locations and a second mask removing the skew-hive seam where \(i+j=n-1\). For every training step, gather the neighboring entries required by the three residuals in the formulas, compute squared ReLU violations, average over valid rhombi and heads, and add \(\lambda\mathcal L_{\mathrm{hive}}\) to the task loss. Start with \(\lambda=10^{-3}\), linearly warm it to \(10^{-2}\) over 10% of training, and compare against isotropic Laplacian smoothness and ordinary total-variation penalties. The mathematics supplies the exact local inequalities and seam exception; the tensor layout, coefficient \(\lambda\), and choice of score surface are empirical. For a cheap experiment, train a 6-layer ViT-Tiny on CIFAR-100 or ImageNet-100 with identical seeds and budgets, comparing baseline, Laplacian regularization, and hive regularization. Track validation accuracy, attention entropy, fraction of violated rhombi, and loss/gradient variance. Success means fewer violations, smoother and more stable training, and at least a small accuracy or calibration gain without materially increasing FLOPs; failure is unchanged accuracy or attention collapse toward uniform scores.