Distance preservers for Lobachevsky space

What the math gives to ML

The paper gives a complete, constructive characterization of scalar functions that preserve Lorentz–Gram matrices under entrywise application. A Lorentz–Gram matrix encodes pairwise inner products of points on the future unit hyperboloid, so the result supplies a principled family of nonlinear transformations of hyperbolic similarities that cannot leave the realizable hyperbolic geometry. The transferable asset is the Lévy–Khintchine/Bernstein-function parameterization: instead of applying an unconstrained learned activation to hyperbolic pairwise scores, a network can use a small set of positive parameters that guarantees geometric validity. This is most naturally tested as a hyperbolic attention, retrieval, or graph-message-passing module with a Lorentz-Gram-preserving similarity transform.