{
 "artifacts": null,
 "category": "architecture",
 "description": "Construct positional or relative-position features as a nonnegative mixture of lattice cosine functions instead of independently signed sinusoidal features. The resulting bias is the Fourier transform of a positive discrete measure with explicitly bounded spectral support, while the mesh and degree can be initialized in the paper's dense-but-controlled frequency regime.",
 "formulas_latex": [
  "$$Q(x)=\\sum_{j=0}^{r}a_jT_j(x),\\qquad a_j\\ge 0,\\qquad T_j(\\cos t)=\\cos(jt).$$",
  "$$f(z)=C Q(\\cos(dz))=C\\sum_{j=0}^{r}a_j\\cos(jdz)=\\int_{\\mathbb R}e^{iz\\xi}\\,d\\mu(\\xi),$$\n$$\\mu=C a_0\\delta_0+\\frac{C}{2}\\sum_{j=1}^{r}a_j(\\delta_{jd}+\\delta_{-jd}),\\qquad \\operatorname{supp}(\\mu)\\subseteq[-rd,rd].$$",
  "$$\\limsup_{k\\to\\infty}r_kd_k^2\\le4\\sum_{\\gamma_n\u003eA}\\frac{1}{\\gamma_n^2}.$$",
  "$$d^{-1}=o(r),\\qquad r=o(d^{-2}),\\qquad\\text{equivalently}\\qquad rd\\to\\infty,\\quad rd^2\\to0.$$"
 ],
 "id": 2790,
 "implementation": "1. Integration point: replace a standard relative-position Fourier bias or random Fourier-feature similarity inside a small Transformer attention block. For token positions u and v, define Delta = u - v and compute b(Delta) = sum_j p_j cos(j d Delta), then add alpha b(Delta) to the attention logits, where alpha is a learned scalar or per-head scale. 2. Pseudocode: choose degree r and initialize d = r^{-beta} with 0.5 \u003c beta \u003c 1; initialize trainable logits theta in R^(r+1); on every forward pass compute p = softmax(theta), construct B[u,v] = sum_{j=0}^r p[j] cos(j*d*(u-v)), and use softmax(QK^T/sqrt(head_dim) + alpha*B). Optionally learn log d, but constrain it or add penalties lambda_1 max(0, epsilon - r*d)^2 + lambda_2 max(0, r*d^2 - eta)^2. 3. The exact mathematical structure computed from the paper is coefficient positivity and spectral support bounded by r*d. The quantities estimated empirically are the effective bandwidth and concentration of p; define B95 = d times the smallest m such that sum from j = 0 to m of p_j is at least 0.95. 4. First experiment: train a 2-layer, 4-head Transformer on a synthetic copy/reversal task and a small character-level WikiText-2 or enwik8 subset. Compare against sinusoidal positions, random Fourier features, and learned relative-position bias at matched parameter count and FLOPs. Test r in {32, 64, 128} and d = r^{-0.75}. Measure validation loss, extrapolation at twice the training sequence length, gradient-norm variance, and attention-logit overflow. The hypothesis is improved length extrapolation and lower optimization variance at equal compute; reject the method if it fails to beat sinusoidal features on extrapolation or stability.",
 "math_summary": "Let Q(x) = sum from j = 0 to r of a_j T_j(x), where T_j is the Chebyshev polynomial, r is the degree, and a_j \u003e= 0. Since T_j(cos(t)) = cos(jt), f(z) = C Q(cos(dz)) = C sum from j = 0 to r of a_j cos(jdz), where C \u003e 0 is a scale and d \u003e 0 is the frequency mesh. This is the Fourier transform of the positive discrete measure mu = C a_0 delta_0 + (C/2) sum from j = 1 to r of a_j(delta_{jd} + delta_{-jd}), whose support lies in [-rd, rd]. The paper proves that under local uniform convergence to an entire F of order below two, with F(0) \u003e 0 and the relevant polynomial zeros in [-1,1), limsup r_k d_k^2 \u003c= 4 sum over gamma_n \u003e A of 1/gamma_n^2, where plus or minus gamma_n are the nonzero real zeros of F. Because the reciprocal-square tail tends to zero, the resulting asymptotic design window is r d -\u003e infinity and r d^2 -\u003e 0, equivalently d^{-1} = o(r) and r = o(d^{-2}). For a neural implementation, normalize the coefficients as p_j = exp(theta_j) / sum_i exp(theta_i), making the learned bias b(Delta) = sum from j = 0 to r of p_j cos(j d Delta) a positive cosine expectation.",
 "math_tags": [
  "harmonic-analysis",
  "measure-theory",
  "approximation-theory",
  "probability"
 ],
 "ml_areas": [
  "attention",
  "embedding",
  "transformer",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.28792",
  "arxiv_url": "https://arxiv.org/abs/2608.28792",
  "summary_what_math_gives_to_ml": "The paper gives a constructive representation of functions as positive cosine mixtures on a frequency lattice, together with a scaling law linking lattice spacing, polynomial degree, and spectral bandwidth. The transferable asset is the identity Q(cos(dz)) = sum_j a_j cos(jdz) with nonnegative coefficients, which makes the function the Fourier transform of a positive discrete measure with bounded support. In neural networks, this suggests replacing unconstrained Fourier features or oscillatory positional biases with positive lattice spectral mixtures. The paper's asymptotic window provides a principled initialization and regularization regime, although its approximation guarantees must be validated empirically in neural models.",
  "title": "Mesh-Degree Rigidity for Positive Chebyshev-Fourier Approximants",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 7,
  "usefulness": 5
 },
 "solves": [
  "stability",
  "accuracy",
  "sample-efficiency"
 ],
 "title": "Positive Lattice Fourier Features",
 "url": "https://synthcore.org/idea/2790/positive-lattice-fourier-features",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}