Stability of the local-to-global construction

Derive quantitative stability bounds for global Lorentz-Gram realizability and for the clique-product completion when prescribed clique matrices have local defects and separator matrices are uniformly well conditioned, with bounds expressed in terms of those defects, separator condition numbers, and clique-tree geometry.

Background

The paper establishes an exact local-to-global completion theorem for chordal graphs and gives explicit clique-product formulas involving inverses of separator matrices. These exact formulas suggest sensitivity to perturbations in the local clique data and to ill-conditioning of separator blocks, but no quantitative perturbation theory is developed. The open problem is to control both the correction needed to obtain globally realizable data and the resulting completion error from local defects and structural conditioning parameters.

References

Suppose that the prescribed clique matrices are close to Lorentz-Gram matrices and that the separator matrices are uniformly well conditioned. Can one bound the perturbation required to obtain globally realizable data in terms of the local defects? Likewise, can the sensitivity of the clique-product completion be controlled in terms of the separator condition numbers and the geometry of the clique-tree?

— Hyperbolic distance matrix completion  (2609.10403 - Putinar et al., 9 Sep 2026) in Section ‘Open questions’, Question ‘Stability of the local-to-global construction’