Regularity-Matched Random Fourier Layer

1. Integration point: use this as the frozen input projection in a coordinate MLP, neural operator, PINN, or implicit neural representation. Given inputs X in R^(B by d), sample N frequencies once at initialization, compute Z_bn = sqrt(2) cos(X_b dot w_n + b_n), and feed Z to a trainable linear or MLP head. 2. Pseudocode: choose a target regularity mode. For Sobolev, sample each w_n from the normalized density proportional to (1+||w||^2)^(-bar{s}); for Gevrey, sample from the density proportional to exp(-2 bar{kappa} ||w||^(1/s)); for ultra-analytic, sample from exp(-2 bar{kappa} ||w||^s). Draw b_n uniformly from [0, 2 pi], form Z, estimate the empirical Gram matrix G = Z^T Z / B + epsilon I, and replace Z by Z G^(-1/2) using an eigendecomposition or QR factorization. Train the downstream head first and optionally unfreeze frequencies later. 3. Compute the distributions and L_N exactly from the paper; estimate the Gram spectrum empirically and choose epsilon = 10^(-5) trace(G)/N. 4. First experiment: fit 2D Sobolev, Gevrey, and bandlimited synthetic functions on a 64 by 64 grid using a linear head with 256, 512, and 1024 features. Compare Gaussian RFF, uniform-window RFF, and matched distributions at equal feature count and FLOPs, with and without whitening. Measure test L2 error, convergence slope versus N, condition number, and wall-clock stability. Success means lower error at fixed N, a steeper log-error curve for smooth targets, and no instability after whitening.