Screened Cross-Correlation Regularizer

1. Integration point: apply the regularizer to hidden states indexed by sequence position, spatial pixel, graph distance, or diffusion time. Generate \(K\) stochastic replicas of a network using dropout, injected Gaussian noise, or stochastic depth. For each pair of positions \(i,j\), compute \(C_{ij}=K^{-1}\sum_k(h_{F,i}^k-\bar h_{F,i})\cdot(h_{B,j}^k-\bar h_{B,j})/d_h\), where \(d_h\) is hidden width. 2. Pseudocode: estimate \(C\) on a minibatch; compute relative-distance bins \(r=|i-j|\); form a binned covariance \(C_r\); evaluate the discrete residual \(R_r=D_{eff}(C_{r+1}-2C_r+C_{r-1})-\mu_{FB}C_r\); add \(\lambda_s\sum_r(R_r-\widehat S_r)^2\) to the task loss. In a first implementation set \(\widehat S_r=0\) for \(r>r_0\), and use \(D_{eff}=1\), \(\mu_{FB}=1/\xi^2\) as a hyperparameter. Estimate \(\xi\) jointly by fitting the empirical tail to \(A\exp(-r/\xi)\), or schedule it from a desired receptive-field scale. 3. Computed from the paper: the screened operator, positivity requirement for \(\mu_{FB}\), and the length-scale relation. Estimated empirically: the source term, covariance, finite-difference Laplacian, and fitted decay length. Use stop-gradient through the covariance estimate initially to avoid second-order stochastic gradients. 4. First cheap experiment: train a small Transformer on synthetic copy and long-range parity tasks, comparing no covariance regularizer, ordinary decorrelation, and screened regularization at matched parameter count. Sweep \(\mu_{FB}\) and verify the predicted boundary: when fitted \(\xi\) is much smaller than the task dependency length, accuracy should drop; when \(\xi\) exceeds that length, the regularizer should have little effect. For local tasks, the measured covariance tail should fit \(e^{-r/\xi}\) with fitted \(\xi\) within 20% of \(\sqrt{D_{eff}/\mu_{FB}}\). Increasing \(\mu_{FB}\) should shorten the correlation range and reduce batch-to-batch hidden-state variance.