Absolute-Convex-Hull Diversity Regularizer

(1) Integration point: apply the regularizer to a matrix V in R^{n x k}, whose row v_i is a learned class prototype, expert key, output embedding, or a minibatch of normalized hidden features. Add lambda times R to the task loss during training only. For large n, compute the statistic on a random subset of vectors or maintain an exponential moving average of S. (2) Pseudocode: V = row_normalize(V_raw); S = (V.T @ V) / n; Sbar = k * S / (trace(S) + eps); A = Sbar + eps * eye(k); Ldiv = -slogdet(A); loss = task_loss + lambda * Ldiv; loss.backward(); optimizer step. Use a Cholesky-based log determinant when A is positive definite. If vector norms carry useful information, omit row normalization but retain trace normalization. (3) Computed from the paper's mathematics: the empirical frame operator S, its trace-normalized version Sbar, its determinant, and its eigenvalue spectrum. The convex-hull volume is not estimated directly. Empirically monitor the smallest eigenvalue, condition number, effective rank, task loss, and pairwise cosine similarity. Compare against no regularizer and the standard pairwise penalty ||V V^T - I||_F^2, using matched regularization coefficients and wall-clock accounting. (4) First experiment: train a small CIFAR-10 ResNet with a 128-dimensional normalized prototype classifier, or train a 10k-class linear classifier with 128-dimensional class prototypes. Sweep lambda over {1e-4, 1e-3, 1e-2, 1e-1} with identical seeds, optimizer, and augmentation. Measure test accuracy, calibration, few-shot accuracy, minimum eigenvalue of Sbar, effective rank exp(-sum_j p_j log p_j) where p_j=lambda_j/sum_l lambda_l, and regularizer overhead. A successful result is higher effective rank and minimum eigenvalue with equal or better accuracy and no material slowdown. A secondary MoE experiment should regularize expert key vectors and measure router-load entropy and the fraction of experts receiving negligible traffic.