Tightness of the structured-inverse landscape condition

Determine whether condition k − ℓ > (1 − η)(min{ℓ, p − k} + 2) in the structured-inverse landscape theorem is tight over the full class of measurement operators satisfying the frame conditions (F1)–(F3).

Background

The paper proves benignness for the factorized positive-semidefinite optimization problem under a structured-inverse hypothesis in which the inverse measurement operator has the form L⁻¹ = I − Γ and Γ is generated by frame atoms satisfying conditions (F1)–(F3). The resulting sufficient inequality controls the gap between the factorization dimension k and the ground-truth rank bound ℓ.

The authors explicitly note that the condition is not tight in the codimension-one case m = p − k = 1, and they do not expect tightness for the complete-graph Euclidean distance operator. Whether the condition is nevertheless sharp for some operators in the full admissible family is left unresolved.