# Entropy-calibrated hyperbolic curvature

- ID: 3051
- Canonical URL: https://synthcore.org/idea/3051/entropy-calibrated-hyperbolic-curvature
- API JSON: https://synthcore.org/api/idea/3051.json
- API Markdown: https://synthcore.org/api/idea/3051.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2609.02362](https://arxiv.org/abs/2609.02362)
- Category: geometry
- Solves: accuracy, stability, sample-efficiency
- ML areas: embedding, graph-nn, training-dynamics, optimizer
- Math tags: information-theory, probability, metric-geometry, differential-geometry
- Ratings: usefulness 7/10; difficulty 6/10; novelty 7/10

## Idea description

Use farthest-triplet entropy as a low-bandwidth observable of whether a hyperbolic embedding is angular- or radial-dominated, then adapt the hyperbolic curvature rather than fixing it arbitrarily. In the isotropic shell regime, invert the entropy-to-signal curve to estimate the effective radial/angular parameter and select curvature that reaches a chosen geometric operating point.

## Mathematical statement

Let X_d=R_d U_d be an isotropic d-dimensional shell, with tau_d=sd(R_d)/E[R_d]. In the Euclidean high-dimensional approximation, farthest ordering is represented by scores S_i(lambda)=lambda xi_i-Z_i, where xi_i are approximately standard Gaussian angular fluctuations, Z_i are independent radial/noise variables, and lambda_d=sqrt(d) tau_d. Define H_infinity(lambda;F) as the expected entropy of the winning index among three independent scores, with F denoting the calibrated law of Z_i. In hyperbolic space of curvature -kappa_d^2, let s_d=kappa_d E[R_d] and A(s)=s coth(s). The paper gives the effective parameter lambda_d,H=sqrt(d) tau_d A(s_d). Therefore, after estimating observed entropy H_hat, invert the monotone master curve to obtain lambda_hat=H_infinity^{-1}(H_hat;F), then solve lambda_hat=sqrt(d) tau_hat s coth(s) for s and set kappa_hat=s/E_hat[R]. This is model-calibrated and should not be trusted under strong angular anisotropy.

## Key formulas

- $$\lambda_d=\sqrt d\,\frac{\operatorname{sd}(R_d)}{\mathbb{E}R_d}=\sqrt d\,\tau_d,$$
- $$S_i(\lambda)=\lambda\xi_i-Z_i,\qquad H_\infty(\lambda;F)=\mathbb{E}\left[-\sum_{i=1}^{3}p_i(\lambda)\log p_i(\lambda)\right],$$
- $$\lambda_{d,\mathbb H}=\sqrt d\,\tau_d A(s_d),\qquad s_d=\kappa_d\mathbb{E}R_d,\qquad A(s)=s\coth s,$$
- $$\hat H\xrightarrow{H_\infty^{-1}}\hat\lambda,\qquad \hat\lambda=\sqrt d\,\hat\tau\,\hat s\coth(\hat s),\qquad \hat\kappa=\frac{\hat s}{\hat\mu_R}.$$

## Implementation notes

Integrate this as an outer-loop controller for a hyperbolic embedding model, such as a Poincare-ball encoder, hyperbolic prototypical classifier, or hyperbolic graph neural network. Every K optimizer steps, draw query embeddings z_b and three prototype embeddings, compute hard farthest labels using the hyperbolic distance, and estimate H_hat from empirical cell frequencies with Laplace smoothing. Estimate dimension d from the embedding width and estimate mu_hat_R and tau_hat=std(R)/mean(R) from current prototype or query radii; use exponential moving averages to suppress noise. Before training, numerically tabulate H_infinity(lambda;F) by Monte Carlo: sample three independent standard-normal angular variables xi_i and three independent radial variables Z_i from the chosen calibrated law F, select argmax_i(lambda xi_i-Z_i), and interpolate entropy over a grid of lambda. Invert the table only on a validated monotone interval. Solve g(s)=sqrt(d) tau_hat s coth(s)-lambda_hat=0 by bisection for s>=0, then update curvature toward kappa_hat=s/mu_hat_R using a slow EMA or optimize log(kappa). The mathematical quantities are the entropy statistic, master curve, and curvature amplification s coth(s); empirical estimates are H_hat, mu_hat_R, tau_hat, and anisotropy diagnostics. First test on synthetic isotropic tree-like data and MNIST or CIFAR-10 hyperbolic classification with fixed-curvature baselines, sweeping initial curvatures. Measure validation accuracy, calibration, hard farthest entropy, estimated-versus-true curvature, and training stability. Success is lower validation loss or better accuracy at equal dimension, convergence to the known synthetic curvature regime, and reduced sensitivity to initialization. Run a separate anisotropic-data test; degradation there is an expected falsifiable failure mode.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: no_effect
- Verdict: Built a NumPy/SciPy MVP for the conditional farthest-triplet entropy master curve, monotone inversion, curvature amplification A(s)=s coth(s), and a slow EMA curvature controller. The core claim was numerically observed: entropy decreased strictly over the tested lambda grid (H(0)=1.0986, H(5)=0.2104), and the controller reduced mean absolute curvature error from 1.3667 for fixed initializations to 0.0204. Anisotropy produced inference errors up to 0.219, supporting the stated robustness limitation; the apparent controller win is synthetic and model-consistent rather than evidence from a trained neural embedding.

### Mechanism check

- Verdict: Built a NumPy/SciPy MVP for the conditional farthest-triplet entropy master curve, monotone inversion, curvature amplification A(s)=s coth(s), and a slow EMA curvature controller. The core claim was numerically observed: entropy decreased strictly over the tested lambda grid (H(0)=1.0986, H(5)=0.2104), and the controller reduced mean absolute curvature error from 1.3667 for fixed initializations to 0.0204. Anisotropy produced inference errors up to 0.219, supporting the stated robustness limitation; the apparent controller win is synthetic and model-consistent rather than evidence from a trained neural embedding.
- Confidence: 7/10
- Limitations: No trained Poincare-ball encoder, MNIST/CIFAR classification, validation accuracy, calibration, wall-clock comparison, or real hyperbolic-distance experiment was tested. The master curve assumes a standard-normal calibrated radial law and isotropic angular behavior; the controller target entropy is generated from the same assumed model, so the recovery result does not establish an end-to-end ML improvement.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.05505
- Paired wins: 7/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_entropy_curvature.py](https://synthcore.org/code/1226/bench_entropy_curvature.py)
- [bench_report.json](https://synthcore.org/code/1226/bench_report.json)
- [entropy_curvature.py](https://synthcore.org/code/1226/entropy_curvature.py)
- [report.md](https://synthcore.org/code/1226/report.md)
- [report_bench_2026-09-03T130830.md](https://synthcore.org/code/1226/report_bench_2026-09-03T130830.md)
- [results.json](https://synthcore.org/code/1226/results.json)
- [Download all files as ZIP](https://synthcore.org/download/1226)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.