Fourier Anti-Concentration Regularizer

Integrate the penalty after the encoder projection and before the task head, or on latent vectors of a VAE or diffusion autoencoder. First apply batch-wise centering and whitening, then sample 8–32 unit projection vectors \(u\), and evaluate frequencies \(\Xi=\{1,2,4,8,16,32\}\). Pseudocode: compute projected values \(z_{u,j}=u^\top h_j\); for every projection and frequency calculate real and imaginary batch means of \(\exp(-2\pi i\xi z_{u,j})\); take their magnitude; add the hinge penalty in the formula to the task loss; backpropagate; and update \(C_u\) using an exponential moving maximum so the regularizer does not impose an arbitrary absolute scale. Estimate \(\widehat\delta\) every 100 steps using a correlation-dimension fit: count pairs with \(|u^\top h_i-u^\top h_j|<r\), fit the slope of \(\log N(r)\) versus \(\log r\), and average across projections. The paper supplies the explicit exponent and threshold \(\delta=1/2\); the dimension estimator, projections, frequency grid, and constants are empirical adaptations. The first cheap experiment is a small VAE on MNIST or CIFAR-10 with standard reconstruction plus KL loss, comparing no penalty, fixed \(\alpha=0.05\), and adaptive \(\alpha(\widehat\delta)\). Evaluate reconstruction, latent collapse, and held-out Fourier slopes. The quantitative prediction is that runs with measured \(\widehat\delta>0.5\) should show a positive log-log decay slope near \(-\alpha(\widehat\delta)\), whereas runs below 0.5 should not show a stable positive-theory slope; the penalty should reduce high-frequency characteristic-function plateaus without increasing latent variance collapse.