{
 "artifacts": null,
 "category": "training",
 "description": "For bandlimited or progressively higher-frequency targets, construct a nested random Fourier layer whose frequency window grows according to the paper's bandwidth laws instead of sampling all frequencies from one fixed range. Train with a low-bandwidth subset first, then activate additional frequency blocks. This creates a mathematically motivated spectral curriculum and may reduce early optimization difficulty while preserving high-frequency accuracy.",
 "formulas_latex": [
  "$$\\lambda_{J,S}=C_{a,d}\\left(\\frac{\\pi}{S}\\right)^{da}J^{-1}(1+SR)^{a}\\left(\\frac{SR}{J}\\right)^{2aJ}$$",
  "$$N\\ge 6J^{d}\\log\\left(\\frac{28J^{d}}{\\delta}\\right)$$",
  "$$S_J=\\frac{(J\\log J)^{1/s}}{4R_*}$$",
  "$$\\widehat U(\\xi)=0\\quad\\text{for }\\xi\\notin Q_S,\\qquad Q_S=[-S,S]^d$$"
 ],
 "id": 3105,
 "implementation": "1. Integration point: modify the Fourier-feature embedding before a coordinate MLP or neural operator. Partition frequencies into nested blocks W_1 subset W_2 subset ... , where block k contains frequencies sampled uniformly from Q_(S_k) = [-S_k,S_k]^d. Keep all blocks in memory but gate later blocks off during early training. 2. Pseudocode: estimate the domain radius R, choose delta = 0.05, and select orders J_k = 2^k J_0. Set S_k = (J_k log J_k)^(1/s) / (4 R_*) for ultra-analytic data, or use a known terminal bandwidth S for bandlimited data. Allocate N_k features so cumulative count satisfies N_k \u003e= 6 J_k^d log(28 J_k^d / delta). At epoch e, activate the smallest k allowed by the curriculum schedule, compute Z_k = sqrt(2) cos(X W_k + b_k), concatenate active blocks, and rescale each block by its running RMS. 3. Compute S_k, J_k, and the sample-size targets from the displayed formulas. Estimate unknown R_*, s, and terminal bandwidth from a validation Fourier spectrum. Use lambda_(J,S) as a conditioning diagnostic; when it becomes very small, apply ridge stabilization to the active feature head using G + rho I, with rho proportional to lambda_(J,S) after clipping to a numerically safe interval. 4. First cheap experiment: regress 2D synthetic functions with known Fourier bandwidth on a small MLP using 64 to 1024 total features. Compare the nested curriculum against all features active from step zero and fixed-window RFF at equal total FLOPs. Track loss during the first 20 percent of updates, final test error, gradient-norm spikes, and activation-block utilization. Success means faster early loss descent, fewer optimizer instabilities, and lower high-frequency test error at the same final feature count.",
 "math_summary": "Let U be an L2 function on R^d whose Fourier transform is supported in the cube Q_S, let the spatial domain Omega lie inside a translated cube Q_R, and let J \u003e= max{2 S R, 2} be an integer approximation order. The paper defines the leverage quantity lambda_(J,S) = C_(a,d) (pi/S)^(d a) J^(-1) (1+S R)^a (S R/J)^(2 a J), where a is in (0,1), C_(a,d) \u003e= 1 depends on a and d, S is the target bandwidth, and R is the spatial radius. It requires N \u003e= 6 J^d log(28 J^d / delta), where N is the number of sampled features and delta is the failure probability. For an ultra-analytic schedule, the paper gives S_J = (J log J)^(1/s) / (4 R_*), where s \u003e 1 is a regularity parameter and R_* is a domain-dependent radius. For a bandlimited target, use the terminal bandwidth S directly. Adapt these results into nested windows Q_(S_1) subset Q_(S_2) subset ... .",
 "math_tags": [
  "approximation-theory",
  "spectral-theory",
  "probability",
  "linear-algebra"
 ],
 "ml_areas": [
  "embedding",
  "mlp",
  "curriculum",
  "training",
  "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": 5,
  "novelty": 6,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "sample-efficiency",
  "stability"
 ],
 "title": "Nested Bandwidth Feature Curriculum",
 "url": "https://synthcore.org/idea/3105/nested-bandwidth-feature-curriculum",
 "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
   }
  }
 }
}