Effective-Radius Calibration for Hyperbolic Embeddings

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.