Stabilize higher-derivative approaches for uniform Cartesian sampling via the covering map

Develop a reliable and accurate method that uses the Hessian and higher derivatives of the covering map π_G (from Cayley coordinates to Cartesian configurations) to adjust Cayley step sizes and achieve uniform Cartesian sampling of the branched covering space R for distance-constrained configuration spaces, overcoming linearization error and ill-conditioning associated with pseudoinverses of the Jacobian/Hessian.

Background

The paper discusses a standard workaround for achieving uniform Cartesian sampling: traverse the Cayley base space while adjusting step sizes using pseudoinverses of the Jacobian of the covering map. This approach suffers from inaccuracies due to linearization and ill-conditioning problems.

A commonly proposed improvement is to use the Hessian and higher derivatives of the covering map. The authors note that such efforts are ongoing and explicitly state that the problem is not settled, highlighting the need for stabilizing derivative-based methods that avoid ill-conditioning while delivering accurate sampling or volume computation.