{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace the usual isotropic Gaussian random Fourier features with a frequency distribution matched to the expected spectral regularity of the target function. For coordinate fields, operator-learning maps, or PDE solution surrogates, this should place more features where the target Fourier energy lies and improve approximation at the same feature count. Stabilize the resulting feature matrix with whitening or ridge regression because spectral accuracy can create severe ill-conditioning.",
 "formulas_latex": [
  "$$L_N := \\frac{N}{\\log(N/\\delta)}$$",
  "$$\\mathrm{d}\\tau_{\\mathrm{Sob}}(w) \\propto (1+|w|^{2})^{-\\bar{s}}\\mathrm{d}w, \\qquad \\|f-f_N\\| \\text{ rate} \\sim L_N^{-(s-t)/d}$$",
  "$$\\mathrm{d}\\tau_{\\mathrm{Gevrey}}(w) \\propto \\exp(-2\\bar{\\kappa}|w|^{1/s})\\mathrm{d}w, \\qquad \\|f-f_N\\| \\text{ rate} \\sim \\exp(-cL_N^{1/(sd)})$$",
  "$$\\mathrm{d}\\tau_{\\mathrm{ultra}}(w) \\propto \\exp(-2\\bar{\\kappa}|w|^{s})\\mathrm{d}w, \\qquad \\|f-f_N\\| \\text{ rate} \\sim \\exp(-cL_N^{1/d}\\log L_N)$$"
 ],
 "id": 3104,
 "implementation": "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.",
 "math_summary": "Let N be the number of sampled features, delta in (0,1) the failure probability, d the input dimension, and L_N := N / log(N/delta). The paper's rate table states that Sobolev targets of smoothness s, measured in an error norm indexed by t, use d tau_Sob(w) proportional to (1+|w|^2)^(-bar{s}) d w and achieve rate L_N^(-(s-t)/d). Gevrey targets with parameter s \u003e= 1 use d tau_Gevrey(w) proportional to exp(-2 bar{kappa}|w|^(1/s)) d w and achieve exp(-c L_N^(1/(s d))). Ultra-analytic targets with s \u003e 1 use d tau_ultra(w) proportional to exp(-2 bar{kappa}|w|^s) d w and achieve exp(-c L_N^(1/d) log L_N). Here bar{s}, bar{kappa}, and c are positive constants determined by the distribution and target class. Implement phi_w(x) = sqrt(2) cos(w^T x + b), with b uniform on [0,2 pi], and train a head on the resulting feature matrix. The extracted text states that the same spectral mechanism causing high accuracy also causes severe condition-number growth, so empirical whitening is part of the adaptation.",
 "math_tags": [
  "approximation-theory",
  "spectral-theory",
  "probability",
  "linear-algebra"
 ],
 "ml_areas": [
  "embedding",
  "mlp",
  "initialization",
  "neural-operator"
 ],
 "paper": {
  "arxiv_id": "2609.03401",
  "arxiv_url": "https://arxiv.org/abs/2609.03401",
  "summary_what_math_gives_to_ml": "The paper provides a principled method for choosing random Fourier feature distributions from the expected regularity of the target, rather than using a fixed Gaussian or uniform frequency law. Its transferable asset is spectral approximation: Sobolev targets obtain algebraic rates, Gevrey targets stretched-exponential rates, and ultra-analytic or bandlimited targets super-exponential rates as the feature count grows. This can become a frozen, regularity-adapted input layer followed by a trainable linear or shallow nonlinear head. Because the paper also identifies severe ill-conditioning as the price of spectral accuracy, implementations should combine the feature distribution with whitening or ridge stabilization.",
  "title": "Spectral Convergence of Random Feature Method in Multiple Dimensions",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 5,
  "usefulness": 7
 },
 "solves": [
  "accuracy",
  "sample-efficiency",
  "stability"
 ],
 "title": "Regularity-Matched Random Fourier Layer",
 "url": "https://synthcore.org/idea/3104/regularity-matched-random-fourier-layer",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}