Characterization of the intermediate curvature-error regime

Characterize the relative Gauss–Newton curvature-error values and Hessian behaviors that can occur between the accurate-minimizer regime with relative error below [...] and the indefinite-Hessian regime with relative error at least 15/8 within the low-cost sets considered for ridge-regularized nonlinear least-squares objectives.

Background

The uniform coexistence theorem establishes two separated regimes in the same low-cost set: every global minimizer has relative Hessian error below c_*=(1+sqrt(2))/8, while another point has an indefinite complete Hessian and relative error at least 15/8. The result does not determine what happens between these two bounds.

The authors explicitly state that the intermediate values are not classified. Resolving this would require determining whether all intermediate error levels can occur, identifying restrictions on the attainable values, or establishing a sharper transition description for the relative Gauss–Newton approximation error.