{
 "artifacts": null,
 "category": "optimization",
 "description": "Use the paper's certified-well geometry to turn continuous localization into a cheap grid proposal stage followed by fixed-step refinement. Threshold the projection-residual score on a coarse grid, then run a bandwidth-calibrated gradient map only from accepted points and merge converged duplicates. This avoids dense optimization from every possible coordinate and is suitable for neural slot or source heads that must return a variable number of continuous locations.",
 "formulas_latex": [
  "$$\\widetilde q(\\bm z)=1-\\|P_{\\widetilde{\\mathcal U}}\\varphi_{\\bm z}\\|_2^2,$$",
  "$$\\bm z_{t+1}=\\bm z_t-h\\nabla\\widetilde q(\\bm z_t),\\qquad h\\asymp\\kappa^{-2},$$",
  "$$\\widetilde q\\text{ has a unique strongly convex minimizer in each }B_{\\gamma/\\kappa}(\\bm x_j),$$",
  "$$\\kappa\\delta_X\\gtrsim s^{2/3}.$$"
 ],
 "id": 3138,
 "implementation": "Attach the refinement procedure after a neural network that outputs a low-rank feature subspace and needs continuous coordinates. Use a bounded domain \\(D\\subset\\mathbb R^d\\), construct a grid with spacing \\(c_g/\\kappa\\), and evaluate \\(q(z)=1-\\|U_s^*\\varphi_z\\|_2^2\\) in batches. Choose a threshold from synthetic training data, initially setting \\(\\tau\\) halfway between the largest positive score and the smallest negative score; during deployment, calibrate it on a held-out validation set. Accept grid points with \\(q(z)\\leq\\tau\\), suppress points within \\(c_g/(2\\kappa)\\) of a lower-score accepted point, and refine each survivor. Pseudocode: `for z0 in grid: if q(z0)\u003c=tau: z=z0; for t in range(T): z=project_D(z-eta/kappa**2*grad_q(z)); if norm(grad_q(z))\u003ctol: break; append(z,q(z)); merge points within c_merge/kappa`. Start with \\(\\eta=0.1\\) and backtrack if the score increases. For \\(\\varphi_z[m]=e^{i\\kappa\\omega_m^Tz}/\\sqrt M\\), compute \\(\\nabla_z\\varphi_z[m]=i\\kappa\\omega_m\\varphi_z[m]\\) exactly and obtain \\(\\nabla q\\) by autodiff or the analytic projector derivative. The paper determines the bandwidth scaling, basin radius, and separation diagnostic; empirical validation determines \\(\\gamma\\), \\(\\tau\\), and merge radii for learned features. First test on 3D synthetic sources with a small MLP or Transformer encoder, comparing exhaustive gradient descent from every grid point and direct coordinate regression. Measure wall-clock score evaluations, gradient steps, missed sources, duplicate count, and localization error. The expected signal is at least a 2x reduction in refinement evaluations with no increase in missed-source rate, plus stable convergence under subspace perturbations.",
 "math_summary": "The paper states that \\(\\widetilde q\\) has a unique strongly convex well in each ball \\(B_{\\gamma/\\kappa}(\\bm x_j)\\), where \\(\\gamma\\) is a dimensionless basin-radius constant and \\(\\kappa\\) is the wavenumber or feature bandwidth. Outside the union of these balls, the objective has a uniform value gap. If \\(L\\) is a local gradient-Lipschitz constant and \\(\\mu\\) is the local strong-convexity constant, the gradient iteration \\(\\bm z_{t+1}=\\bm z_t-h\\nabla\\widetilde q(\\bm z_t)\\) is invariant in a certified well and converges linearly for a sufficiently small step; the paper identifies the scaling \\(h\\asymp\\kappa^{-2}\\), consistent with \\(L\\asymp\\kappa^2\\). For Fourier features, \\(\\nabla_z\\varphi_z[m]=i\\kappa\\omega_m\\varphi_z[m]\\), so the score gradient is directly computable. The arbitrary-cloud frame analysis gives the sufficient separation scale \\(\\kappa\\delta_X\\gtrsim s^{2/3}\\), where \\(\\delta_X=\\min_{i\\neq j}\\|\\bm x_i-\\bm x_j\\|\\); this is a screening condition for the theorem, not a claim of spectral impossibility below that scale.",
 "math_tags": [
  "geometry",
  "dynamical-systems",
  "optimization",
  "harmonic-analysis",
  "spectral-theory"
 ],
 "ml_areas": [
  "embedding",
  "optimizer",
  "inference-speedup"
 ],
 "paper": {
  "arxiv_id": "2609.03264",
  "arxiv_url": "https://arxiv.org/abs/2609.03264",
  "summary_what_math_gives_to_ml": "The paper develops a robust subspace-matching objective whose minima remain localized under deterministic adversarial perturbations of the estimated signal subspace. Its transferable asset is not spherical MUSIC itself, but the combination of projection-residual scoring, explicit sine-theta robustness, strongly convex local basins, and a fixed step size determined by the spatial bandwidth. This suggests neural modules that localize continuous latent objects from a learned low-rank feature subspace, while providing an adversarially stable score and a cheap coarse-to-fine refinement procedure. The guarantees are most relevant when a network produces Fourier-like or positional embeddings and the downstream task is continuous localization, slot extraction, or source separation.",
  "title": "Certified Spherical MUSIC for 3D Localization under Adversarial Subspace Perturbations",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 8,
  "usefulness": 7
 },
 "solves": [
  "speedup",
  "stability",
  "accuracy"
 ],
 "title": "Certified Coarse-to-Fine Coordinate Refinement",
 "url": "https://synthcore.org/idea/3138/certified-coarse-to-fine-coordinate-refinement",
 "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
   }
  }
 }
}