Directional H-distribution regularizer

Integrate the regularizer on an intermediate CNN feature map or on a ViT patch-token grid reshaped to a two-dimensional lattice. For each sample b, define u_b as the channel mean of the chosen activation and v_b as the corresponding loss residual or gradient with respect to that activation. Detach u_b if the regularizer should affect only the residual pathway; otherwise allow gradients through both tensors. Normalize u and v over each minibatch to remove the zero-frequency mean. Create R Hann or Gaussian spatial windows w_r and K directional masks. For a two-dimensional feature map, construct the FFT frequency grid, compute theta_xi = xi divided by its magnitude, and set a_k(theta_xi) = exp(kappa times the dot product of theta_xi and q_k), where q_k is the unit vector at angle 2 pi k divided by K. Normalize the masks across k and set the zero-frequency mask to zero. Pseudocode: u = normalize(channel_mean(activation); v = normalize(residual); total = 0; for each window r: U = window[r] times u; V = window[r] times v; F = FFT(V); for each direction k: Z = IFFT(F times mask[k]); C[k] = batch_mean(U times Z); total = total + squared(C[k] minus mean(C)); loss = task_loss + lambda times total. The paper supplies the multiplier pairing; FFT resolution, windows, kappa, lambda, and whether to use gradients or residuals are empirical choices. First run CIFAR-10 with ResNet-18, comparing ordinary weight decay with this loss at lambda values 1e-4, 1e-3, and 1e-2, using R = 4 and K = 8. Log clean accuracy, high-frequency corruption accuracy, gradient-norm variance, and directional concentration defined as maximum over k of squared absolute C_{r,k} divided by the sum over j of squared absolute C_{r,j}. A successful result is lower concentration and gradient variance with unchanged clean accuracy, or improved corruption accuracy at equal training FLOPs.