Radial Hyperbolic Measures: Shell Geometry, Pyramid Limits, and Gaussian Phase Transitions

arXiv:2607.27605 2026 Geometry 1 ideas extracted · analyzed Aug 31, 2026

What the 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.

Ideas from this paper

Unverified 2026

Effective-Radius Calibration for Hyperbolic Embeddings

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.

Useful5/10

Difficulty4/10

Novelty6/10

Geometry Stability Accuracy Scalability

Paper: Radial Hyperbolic Measures: Shell Geometry, Pyramid Limits, and Gaussian Phase Transitions arXiv:2607.27605