Anchored-Box Coverage Regularizer

1. Integration point: attach the regularizer to an encoder output, embedding layer, or generator. Map each representation z in R^m to u=sigmoid((z-mu)/tau) in (0,1)^d, using d=8--32 selected coordinates or a learned random projection. Compute it on each minibatch B={u_i}_{i=1}^N. 2. Pseudocode: sample R random box corners y_r in (0,1)^d and coordinate masks M_r; compute q_r=(1/N) sum_i product_{j in M_r} sigmoid((y_{rj}-u_{ij})/tau), target p_r=product_{j in M_r} y_{rj}, and L_box=(1/R)sum_r w_r(q_r-p_r)^2. Optimize L_total=L_task+lambda L_box. The product over M_r is a soft version of the anchored-box indicator; coordinates outside M_r are unrestricted. Anneal tau from 0.2 to 0.03 and clip gradients. 3. The paper supplies the discrepancy definition, the anchored-box transfer inequality, and the h-product expansion. The implementation estimates the supremum empirically by random corners; report max_r |q_r-p_r| and low-order diagnostics |N^{-1}sum_i phi_S(u_i)| for |S|=1,2,3. Maintain an exponential-memory pool of corners with the largest violations and resample half the corners from that pool. 4. First experiment: train a small VAE or 2-layer MLP classifier on MNIST or CIFAR-10 with a 16-dimensional sigmoid bottleneck. Compare no regularizer, pairwise repulsion, covariance whitening, and L_box at equal FLOPs. Measure nearest-neighbor collapse, occupancy of a random-projection grid, held-out likelihood or classification accuracy, and sampled discrepancy. Success means lower maximum box error and fewer empty latent regions without harming task loss; target at least a 10% reduction in collapse metrics at equal accuracy.