# Effective-Radius Calibration for Hyperbolic Embeddings

- ID: 1497
- Canonical URL: https://synthcore.org/idea/1497/effective-radius-calibration-for-hyperbolic-embeddings
- API JSON: https://synthcore.org/api/idea/1497.json
- API Markdown: https://synthcore.org/api/idea/1497.md
- Verification status: unverified
- Source: [arXiv:2607.27605](https://arxiv.org/abs/2607.27605)
- Category: geometry
- Solves: stability, accuracy, scalability
- ML areas: embedding, loss, transformer
- Math tags: metric-geometry, geometry, probability
- Ratings: usefulness 5/10; difficulty 4/10; novelty 6/10

## Idea 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.

## Mathematical statement

The paper defines the effective radius of a shell of raw hyperbolic radius rho under metric rescaling s_n > 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).

## Key formulas

- $$a_{n,\rho}\coloneqq s_n\log\frac{\sinh\rho}{\sqrt{n}},\qquad a_+\coloneqq\max\{a,0\}.$$
- $$r_{\mathrm{eff}}(z)=s\left[\log\frac{\sinh(\rho(z))}{\sqrt{n}}\right]_+,$$
- $$\mathcal{L}_{\mathrm{eff}}=\frac{1}{B}\sum_{i=1}^{B}\left(r_{\mathrm{eff}}(z_i)-r_0\right)^2,$$
- $$\tau_{\mathrm{eff}}=\tau_0\frac{\operatorname{stopgrad}(\overline r_{\mathrm{eff}}+\epsilon)}{r_0+\epsilon},\qquad \ell_{ij}=-\frac{d_{\mathbb H}(z_i,z_j)}{\tau_{\mathrm{eff}}}.$$

## Implementation notes

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.

## Disclaimer

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 Not peer-reviewed.