Logarithmic-Laplacian Feature Regularizer

Integrate the regularizer after one intermediate feature tensor: use the final convolutional feature map before global pooling, or a transformer hidden-state tensor before its prediction head. Represent the tensor as H with shape [batch, N, d], and assign each position a coordinate x_i: two-dimensional normalized pixel coordinates for CNNs, or one-dimensional normalized token coordinates for the first transformer test. Precompute a sparse pair list containing either all pairs with r_ij <= R or the K nearest positions. Normalize coordinates so the unit-ball cutoff has a controlled meaning, and set epsilon to 1e-3 times the median nonzero pair distance. For every batch item, compute the vectorized update Lhat_i = c_n * sum_j w_ij * (H_i * indicator(r_ij < 1) - H_j)/(r_ij + epsilon)^n + rho_n * H_i. Use w_ij=1 for a uniform grid, or quadrature cell areas for irregular coordinates. Compute R as the mean squared L2 norm of Lhat divided by d, then add lambda R to the task loss. Obtain c_n and rho_n using library Gamma, Digamma, and Euler-constant functions; estimate only R, K, epsilon, lambda, and the insertion layer empirically. The first cheap test should use CIFAR-10 with ResNet-18, comparing cross-entropy alone, a nearest-neighbor graph-Laplacian penalty, and this logarithmic penalty while matching regularizer gradient norms. Measure clean accuracy, CIFAR-10-C corruption accuracy, training stability, feature high-frequency energy, and wall-clock overhead. Success means higher validation or corruption accuracy at matched training accuracy, with less feature collapse than ordinary Laplacian smoothing.