Equivalence of the two different Lorentz distance function

[Zhuo Le Lee]

Dear Mr. Boumal, 

Thank you so much for your work in creating manopt! As I was experiencing numerical instability when calculating the Lorentzian distance between two neighbouring points, I came across your equivalent formulation of this distance function that used arcsinh instead of arccosh (line 135-150 here). I have tested this and it works wonders! 

My aim is to derive a curvature-aware variation of this distance function involving arcsinh, i.e. for the case where the curvature of the hyperboloid does not equal to -1. I have arrived at the following derivation of the distance function, which is however still numerically instable for small distances. 

Could you please elaborate on how you derived the equivalent numerically more stable Lorentz distance function in manopt, and how one could tailor this to include the curvature parameter? I can provide more information regarding the derivations from my side if required. 

Thank you so much for your help in advance!

[Nicolas Boumal]

Hello,

To check (and hence also to adapt) how to go from the acosh formula to the asinh formula, you can use the following formulas on Wikipedia, under "Other Identities":

https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions#Other_identities

Best,

Nicolas