{
 "artifacts": null,
 "category": "geometry",
 "description": "Train a neural implicit occupancy or signed-distance model with Fourier coefficients sampled on a dual lattice, while explicitly preventing spatial aliasing under the corresponding periodic lattice. The spatial reconstruction loss is supplemented by a finite Fourier loss and a penalty for shape-point differences that approach nonzero lattice vectors.",
 "formulas_latex": [
  "$$(\\mathcal K-\\mathcal K)\\cap\\mathcal L=\\{0\\},\\qquad \\mathcal K-\\mathcal K:=\\{x-y:x,y\\in\\mathcal K\\}.$$",
  "$$\\widehat{\\mathbf 1_{\\mathcal P}}(\\xi)=\\widehat{\\mathbf 1_Q}(\\xi)\\quad\\text{for every }\\xi\\in\\mathcal L^*\\ \\Longrightarrow\\ Q=\\mathcal P+\\ell\\text{ for some }\\ell\\in\\mathcal L.$$",
  "$$\\widehat f(\\xi)=\\int_{\\mathbb R^d}f(x)e^{-2\\pi i\\langle\\xi,x\\rangle}\\,dx,\\qquad \\mathcal L^*=\\{\\xi\\in\\mathbb R^d:\\langle\\xi,\\ell\\rangle\\in\\mathbb Z\\ \\forall\\ell\\in\\mathcal L\\}.$$",
  "$$L_{\\mathrm{sparse}}=\\frac{1}{M}\\sum_{i\\ne j}\\exp\\left(-\\frac{\\min_{m\\in\\{-M_0,\\ldots,M_0\\}^d\\setminus\\{0\\}}\\|x_i-x_j-Am\\|_2^2}{\\tau^2}\\right),\\qquad \\mathcal L=A\\mathbb Z^d.$$"
 ],
 "id": 2813,
 "implementation": "Integrate the method into a coordinate MLP or neural implicit occupancy model s_theta(x) in [0,1], trained on 2D silhouettes, 3D occupancy grids, or sampled point clouds. Use a spatial reconstruction loss such as binary cross-entropy or signed-distance regression, then add Fourier supervision. Choose a spatial lattice L = A Z^d, where A is an invertible d by d matrix; the dual lattice is L* = A^{-T} Z^d. For each frequency index k in a small integer cube, set xi_k = A^{-T} k. Given sampled coordinates x_j, predicted occupancies p_j = s_theta(x_j), and target values y_j, estimate p_hat_k = (1/N) sum_j p_j exp(-2 pi i inner_product(xi_k, x_j)) and y_hat_k analogously. Optimize L_total = L_spatial + lambda_F sum_k absolute_value(p_hat_k - y_hat_k)^2 + lambda_S L_sparse. For sampled points belonging to the shape, compute L_sparse using the fourth formula, enumerating nonzero integer vectors m in a bounded range. The theorem supplies the exact no-congruence condition and the translation ambiguity; finite frequencies, soft occupancy, and sampled point pairs are empirical approximations. The first experiment should use a small SIREN or Fourier-feature MLP on MNIST silhouettes or low-resolution ShapeNet shapes. Compare spatial supervision against spatial plus Fourier supervision, with and without the sparse penalty, at equal coordinate-query budgets. Measure IoU, boundary Chamfer distance, robustness when only 10 to 25 percent of coordinates are observed, and the number of Fourier coefficients needed to reach a fixed IoU. The expected signal is improved reconstruction or equal reconstruction with fewer samples, together with fewer periodic wrap-around artifacts.",
 "math_summary": "The paper defines the sparse lattice property for a set K contained in R^d by (K-K) intersect L = {0}, where L is a full-rank spatial lattice and K-K = {x-y : x,y in K}. This means that no two distinct points of K are congruent modulo L. Theorem 1.2 states that if P and Q are connected finite unions of convex bodies satisfying this property and their indicator Fourier transforms agree on every point of the dual lattice L*, then Q = P + ell for some ell in L. Here 1_A is the indicator of set A, the Fourier transform is defined by hat f(xi) = integral of f(x) exp(-2 pi i inner product of xi and x) dx, and L* = {xi : inner product of xi and ell is an integer for every ell in L}. The neural version uses a finite subset of dual-lattice frequencies and estimates the coefficients by Monte Carlo sampling, so the theorem becomes a design principle rather than an exact finite-sample guarantee.",
 "math_tags": [
  "harmonic-analysis",
  "geometry",
  "measure-theory"
 ],
 "ml_areas": [
  "embedding",
  "loss",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.29217",
  "arxiv_url": "https://arxiv.org/abs/2608.29217",
  "summary_what_math_gives_to_ml": "The paper proves an exact identifiability theorem for sets from Fourier samples taken on a dual lattice. For connected finite unions of convex bodies satisfying an alias-free sparse-lattice condition, equality of all sampled indicator transforms determines the set up to a lattice translation. The transferable asset is a mathematically explicit condition preventing periodic aliasing in Fourier-feature representations. This suggests combining finite Fourier supervision for neural implicit shapes with a sparse-lattice regularizer or lattice-selection procedure.",
  "title": "A note on a sparse sampling conjecture",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 5
 },
 "solves": [
  "accuracy",
  "sample-efficiency",
  "stability"
 ],
 "title": "Alias-free lattice Fourier supervision",
 "url": "https://synthcore.org/idea/2813/alias-free-lattice-fourier-supervision",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}