Fractional Hilbert-Scale Neural Regularizer

(1) Integration point: apply the regularizer to the final spatial prediction tensor u produced by a coordinate MLP, U-Net, or CNN. Keep the original data loss L_data = ||K(u)-y||_2^2 and add alpha times the fractional spectral energy. For a periodic grid use FFT; for zero-Dirichlet boundaries use a discrete sine transform. (2) Pseudocode: u = net(input); residual = K(u) - y; U = DST(u) or FFT(u); multipliers = 1 + lambda_grid**s; reg = sum(multipliers * abs(U)2) / number_of_grid_points; loss = mean(residual2) + softplus(a) * reg; optimizer.zero_grad(); loss.backward(); optimizer.step(). Here lambda_grid contains the discrete Laplacian eigenvalues and s can be fixed or parameterized as sigmoid(rho) to constrain it to (0,1). (3) Computed from the paper is the spectral multiplier 1 + lambda_k^s and the exact quadratic form; estimated empirically are alpha and s, selected on validation data. For a smoothing K, compute the diagnostic q_k = (1 + lambda_k^s)^(-1) exp(-2t(1 + lambda_k^s)) and report errors only in modes with q_k above a chosen threshold, or use q_k to identify the unrecoverable bandwidth. (4) First experiment: train a small coordinate MLP on 64 by 64 Gaussian deblurring and denoising problems, comparing s values 0.25, 0.5, 0.75, and 1 against weight decay, total variation, and no output regularization. Use identical parameter counts and optimizer settings. Measure test MSE, edge preservation, Fourier error by frequency band, and variance across noise seeds. The expected signal is lower high-frequency error and improved reconstruction stability at equal training FLOPs; for deblurring, success is lower total MSE than s = 1 at the same noise level without oversmoothing sharp edges.