Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification

What the math gives to ML

The paper introduces a comparison-based statistic for the geometry of learned representations: the conditional entropy of the farthest prototype among three, averaged over queries and prototype triples. Its useful properties are that it needs only argmax comparisons, is invariant to monotone rescaling of distances, and detects whether prototype cells are balanced or dominated by radial variation. In high-dimensional isotropic shells, the statistic is governed by a one-dimensional signal-to-noise parameter, while hyperbolic curvature amplifies radial variation through the explicit factor A(s)=s coth s. The most promising neural uses are a farthest-neighbor geometry regularizer and an entropy-based controller for adaptive hyperbolic curvature, with anisotropy explicitly treated as a failure mode.