{
 "artifacts": [
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1079/report.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1079/results.json"
  },
  {
   "name": "spectrum_experiment.py",
   "url": "https://synthcore.org/code/1079/spectrum_experiment.py"
  }
 ],
 "category": "regularization",
 "description": "Constrain a set of learnable or batch-produced unit-norm embeddings by matching their spherical-harmonic power spectrum to a target spectrum rather than relying only on pairwise Euclidean repulsion. This creates an explicit, tunable mechanism for suppressing low-frequency density fluctuations or enhancing a selected angular frequency, which can improve uniformity and reduce representation collapse on hyperspherical embeddings.",
 "download_zip": "https://synthcore.org/download/1079",
 "formulas_latex": [
  "$$\\rho(u)=\\sum_{i=1}^{N}c_i\\,\\delta(u-u_i),\\qquad u_i\\in\\mathbb S^2,\\qquad \\sum_{i=1}^{N}c_i=N.$$",
  "$$a_{\\ell m}=\\int_{\\mathbb S^2}\\rho(u)Y_{\\ell m}^{*}(u)\\,d\\Omega=\\sum_{i=1}^{N}c_iY_{\\ell m}^{*}(u_i),\\qquad P_\\ell=\\frac{1}{2\\ell+1}\\sum_{m=-\\ell}^{\\ell}|a_{\\ell m}|^2.$$",
  "$$\\mathcal L_{\\mathrm{spec}}=\\lambda\\sum_{\\ell=1}^{L}w_\\ell\\left(P_\\ell-P_\\ell^{\\star}\\right)^2,\\qquad \\mathcal L=\\mathcal L_{\\mathrm{task}}+\\mathcal L_{\\mathrm{spec}}+\\mu\\mathcal L_{\\mathrm{rep}}.$$",
  "$$\\mathcal L_{\\mathrm{rep}}=\\frac{1}{N(N-1)}\\sum_{i\\neq j}\\left[\\max\\left(0,\\,\\cos^{-1}(u_i^{\\mathsf T}u_j)-\\alpha\\right)\\right]^2.$$"
 ],
 "id": 2726,
 "implementation": "Integrate this at the representation layer immediately after producing a batch of vectors $z_i\\in\\mathbb R^3$ or after projecting arbitrary embeddings to three dimensions with a learned linear map. Normalize with $u_i=z_i/(\\|z_i\\|+\\epsilon)$, convert each $u_i$ to $(\\theta_i,\\phi_i)$, and evaluate real or complex spherical harmonics up to degree $L$ using a differentiable library. Pseudocode is: `u = normalize(z); a[l,m] = sum_i(c[i] * conj(Y(l,m,u[i]))); P[l] = mean_m(abs(a[l,m])**2); Lspec = lam * sum_l(w[l]*(P[l]-Ptarget[l])**2); loss = task_loss + Lspec + mu*repulsion(u); backprop(loss)`. For an MVP, use $L=8$, equal weights $c_i=1$, and compute the harmonics directly; later replace the dense evaluation by a nonuniform spherical-harmonic transform when $N$ and $L$ are large. The paper supplies the density-to-harmonic construction and the target-spectrum optimization principle; the neural-network gradient and batching are estimated by automatic differentiation. First test a 3-layer MLP on CIFAR-10 or a contrastive image encoder on CIFAR-10 with a 3D hyperspherical prototype layer. Compare no regularizer, cosine repulsion, and spectral regularization with $P_\\ell^\\star=0$ for $1\\leq\\ell\\leq4$. The quantitative prediction is that the measured batch spectrum satisfies $P_\\ell/P_0$ substantially below the unregularized baseline for low degrees and approaches the target with error decreasing as optimization proceeds. Sweep $\\lambda$ and predict a stability boundary: when the regularizer gradient norm becomes comparable to the task gradient, approximately $\\lambda\\|\\nabla\\mathcal L_{\\mathrm{spec}}\\|/\\|\\nabla\\mathcal L_{\\mathrm{task}}\\|\\approx1$, training should begin to oscillate or classification loss should increase. Also measure angular-count variance in spherical caps; low-mode suppression should reduce large-cap variance relative to random embeddings.",
 "math_summary": "For unit vectors $u_i\\in\\mathbb S^2$ with optional weights $c_i$ satisfying $\\sum_i c_i=N$, define the spherical density $\\rho(u)=\\sum_{i=1}^N c_i\\delta(u-u_i)$. Let $Y_{\\ell m}$ be orthonormal spherical harmonics and define coefficients $a_{\\ell m}=\\sum_i c_iY_{\\ell m}^*(u_i)$. The angular power at degree $\\ell$ is $P_\\ell=(2\\ell+1)^{-1}\\sum_{m=-\\ell}^{\\ell}|a_{\\ell m}|^2$, with monopole $a_{00}=N/\\sqrt{4\\pi}$. Given a target spectrum $P_\\ell^\\star$, optimize the neural loss plus $\\lambda\\sum_{\\ell=1}^{L}w_\\ell(P_\\ell-P_\\ell^\\star)^2$. The key mechanism is that low $P_\\ell$ suppresses large-scale angular density fluctuations, while a positive target at one degree $\\ell_0$ creates a controlled spectral feature. Gradients are obtained by differentiating $Y_{\\ell m}(u_i)$ through the normalized embedding $u_i=z_i/(\\|z_i\\|+\\epsilon)$.",
 "math_tags": [
  "harmonic-analysis",
  "geometry",
  "optimization",
  "dynamical-systems"
 ],
 "ml_areas": [
  "embedding",
  "regularization",
  "optimizer"
 ],
 "paper": {
  "arxiv_id": "2608.24867",
  "arxiv_url": "https://arxiv.org/abs/2608.24867",
  "summary_what_math_gives_to_ml": "The paper provides a constructive spectral optimization mechanism for point sets on the sphere: represent a weighted point density with spherical harmonics, then optimize the harmonic power spectrum to impose target pair correlations. Its transferable asset is a fast differentiable repulsion or spectral-shaping regularizer for neural representations constrained to a hypersphere, such as class prototypes, token embeddings, spherical codebooks, or manifold samples. The most promising implementation is to penalize selected spherical-harmonic modes of normalized embeddings while retaining a low-cost real-space repulsion term, with a measurable prediction that the empirical angular power spectrum follows the prescribed target and that low-mode suppression produces hyperuniform-like variance scaling.",
  "title": "Fast generation of spectrally-shaped disorder, on the sphere",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "generalization",
  "stability",
  "accuracy"
 ],
 "title": "Spherical harmonic spectrum regularizer",
 "url": "https://synthcore.org/idea/2726/spherical-harmonic-spectrum-regularizer",
 "verification": {
  "peer_reviewed": false,
  "stage1_mechanism_check": {
   "worked": false,
   "confidence": 8,
   "verdict": "Built a differentiable spherical-harmonic spectrum regularizer using the Legendre addition theorem, with autodiff and optimization sweeps. The core math passed: the addition-theorem l=1 error was 1.1e-16 and the finite-difference/autodiff gradient relative error was 1.35e-8. The mechanism produced a real monotonic signal: q1 fell from 1.000 at lambda=0 to 0.391 at lambda=1, while cap-count variance fell from 432.5 to 73.1. However, the strict stage-1 predictions were not all confirmed: the observed transition began near lambda=0.03, while the initial gradient-ratio boundary was near lambda=0.1-1, and the lambda=0.1 convergence sweep moved spectrum loss from 0.974 to 1.031 rather than decreasing toward the zero-spectrum target; therefore worked=false.",
   "metrics": {
    "baseline": "lambda=0: task=-1.0000, normalized spectrum loss=4.0000, q1=1.0000, cap variance=432.538",
    "gradient_scaling": "spectrum/task gradient norm ratio=8.565 at lambda=1, 0.857 at lambda=0.1, 0.0857 at lambda=0.01",
    "idea": "lambda=1: task=-0.6256, normalized spectrum loss=0.1538, q1=0.3914, cap variance=73.128; lambda=0.1 gives task=-0.9065, spectrum loss=1.0307, q1=0.8217, cap variance=268.560",
    "lambda_sweep": "q1: 1.000 at lambda=0, 1.000 at .01, .930 at .03, .822 at .1, .666 at .3, .391 at 1",
    "math_check": "addition-theorem absolute error=1.11e-16; gradient finite-difference relative error=1.35e-8"
   },
   "how_to_run": "python3 spectrum_experiment.py",
   "files": [
    "spectrum_experiment.py",
    "results.json"
   ],
   "limitations": "Only a synthetic north-pole-collapse toy task was tested; no CIFAR-10, neural representation learning, pairwise-repulsion baseline, learned projection, weighted points, selected nonzero target spectrum, GPU benchmark, or FLOP/runtime comparison was performed."
  },
  "status": "mechanism_failed",
  "status_label": "Mechanism failed",
  "updated_at": "2026-09-01T23:52:45",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}