Plucker Compound-Rank Regularizer

1. Integration point: choose a symmetric matrix A_theta from a neural network, such as the symmetrized feature Jacobian A=(J+J^T)/2, a graph-message-passing interaction matrix, or a small hidden-layer covariance operator. Select t=1, 2, or 3 and use a hidden dimension n small enough to enumerate minors initially. 2. Pseudocode: compute A; enumerate every t by t minor to form C; normalize C by its Frobenius norm; compute its leading eigenvector u by one or two power iterations; add lambda_Pl times the squared Frobenius residual between normalized C and uu^T; add lambda_gap times the squared singular values beyond t; update network parameters by backpropagation. Initially stop gradients through u to avoid eigenvector-instability issues. 3. Computed from the paper: the compound construction and rank-one prediction. Estimated empirically: A, its singular values, u, and the scale kappa. For larger n, estimate the loss from random minor subsets. 4. First experiment: train a two-layer MLP on MNIST with a 16-dimensional feature block, set t=3, and compare ordinary weight decay with the compound penalty at matched accuracy. Log the residual r=||C/||C||_F-uu^T||_F and the ratio sigma_4/sigma_3. The quantitative prediction is that increasing lambda_Pl produces a transition where r sharply decreases and sigma_4/sigma_3 becomes small; the observed transition should be reproducible across seeds and occur before severe task-loss degradation.