{
 "artifacts": null,
 "category": "geometry",
 "description": "Replace raw hyperbolic embedding-radius regularization with a dimension-aware effective-radius target. For embeddings concentrated near hyperbolic radius rho in an n-dimensional hyperbolic space, regulate s times log(sinh(rho) / sqrt(n)) rather than rho itself, and use the same quantity to calibrate distance-logit temperature. This should make hyperbolic metric-learning behavior more invariant when embedding dimension, curvature, or model scale changes.",
 "formulas_latex": [
  "$$a_{n,\\rho}\\coloneqq s_n\\log\\frac{\\sinh\\rho}{\\sqrt{n}},\\qquad a_+\\coloneqq\\max\\{a,0\\}.$$",
  "$$r_{\\mathrm{eff}}(z)=s\\left[\\log\\frac{\\sinh(\\rho(z))}{\\sqrt{n}}\\right]_+,$$",
  "$$\\mathcal{L}_{\\mathrm{eff}}=\\frac{1}{B}\\sum_{i=1}^{B}\\left(r_{\\mathrm{eff}}(z_i)-r_0\\right)^2,$$",
  "$$\\tau_{\\mathrm{eff}}=\\tau_0\\frac{\\operatorname{stopgrad}(\\overline r_{\\mathrm{eff}}+\\epsilon)}{r_0+\\epsilon},\\qquad \\ell_{ij}=-\\frac{d_{\\mathbb H}(z_i,z_j)}{\\tau_{\\mathrm{eff}}}.$$"
 ],
 "id": 1497,
 "implementation": "Integrate the method into a hyperbolic embedding or metric-learning model immediately after the exponential-map output and before pairwise distances are passed to a contrastive loss, retrieval objective, or hyperbolic attention module. Let z_i be a batch of embeddings in hyperbolic dimension n, and compute rho_i as the hyperbolic distance from the origin. Then calculate r_eff_i = s * relu(log_sinh(rho_i) - 0.5 * log(n)), where log_sinh is implemented stably as rho + log1p(-exp(-2 rho)) - log(2) for sufficiently large rho, with a small positive clamp near zero. Add lambda_eff times mean((r_eff_i - r0)^2) to the existing training loss. For contrastive logits, compute the detached batch mean r_bar = mean(r_eff_i), set tau_eff = tau0 * (r_bar + epsilon) / (r0 + epsilon), and divide pairwise negative hyperbolic distances by tau_eff. Pseudocode: rho = origin_distance(z); re = s * relu(stable_log_sinh(rho) - 0.5 * log(n)); loss = base_loss + lambda_eff * mean((re-r0)**2); tau = tau0 * (mean(re).detach()+eps)/(r0+eps); logits = -pairwise_hyperbolic_distance(z,z)/tau. The paper supplies the dimension-aware transformation; the engineer estimates rho from the embedding coordinates and selects r0 from the median effective radius of a baseline model. First test a two-layer hyperbolic MLP on CIFAR-100 or a small hierarchical text-classification dataset at dimensions 32, 128, and 512. Compare against raw-radius regularization and no radius control. Track Recall@K, validation accuracy, pairwise-distance variance, embedding-radius histograms, gradient norms, and loss variance. Success means stable distance distributions and comparable or better Recall@K across dimensions, with reduced norm saturation and no increase in training cost beyond one elementwise regularizer.",
 "math_summary": "The paper defines the effective radius of a shell of raw hyperbolic radius rho under metric rescaling s_n \u003e 0 by a_(n,rho) = s_n log(sinh(rho) / sqrt(n)), where n is the hyperbolic dimension. It also defines a_+ = max(a, 0). The square-root dimension term represents the angular concentration scale of positive-mass subsets on a high-dimensional sphere, while sinh(rho) is the hyperbolic expansion factor. The transferable map is therefore r_eff(z) = s [log(sinh(rho(z)) / sqrt(n))]_+, where rho(z) is the hyperbolic distance from the origin and s is the distance scale used by the model. The proposed loss targets r_eff rather than raw rho: L_eff = mean_i (r_eff(z_i) - r_0)^2. Here r_0 is a chosen target effective radius. This construction is useful because equal raw radii at different dimensions need not produce equal separation of typical subsets. The positive part prevents a negative effective separation when sinh(rho) is below sqrt(n).",
 "math_tags": [
  "metric-geometry",
  "geometry",
  "probability"
 ],
 "ml_areas": [
  "embedding",
  "loss",
  "transformer"
 ],
 "paper": {
  "arxiv_id": "2607.27605",
  "arxiv_url": "https://arxiv.org/abs/2607.27605",
  "summary_what_math_gives_to_ml": "The paper identifies a dimension-dependent failure mode in radial hyperbolic geometry: concentrating embeddings at a fixed raw radius does not determine how far positive-mass subsets actually separate. Its effective-radius quantity, s log(sinh(rho) / sqrt(n)), combines hyperbolic radial expansion with the sqrt(n) scale of angular concentration. This suggests dimension-aware calibration for hyperbolic representation learning: control effective radius rather than raw norm, so changing embedding dimension or curvature does not silently cause distance collapse or saturation.",
  "title": "Radial Hyperbolic Measures: Shell Geometry, Pyramid Limits, and Gaussian Phase Transitions",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 6,
  "usefulness": 5
 },
 "solves": [
  "stability",
  "accuracy",
  "scalability"
 ],
 "title": "Effective-Radius Calibration for Hyperbolic Embeddings",
 "url": "https://synthcore.org/idea/1497/effective-radius-calibration-for-hyperbolic-embeddings",
 "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
   }
  }
 }
}