Canonical completion for singular data

Determine whether singular admissible maximal-clique Lorentz-Gram data on a chordal specification graph admit a distinguished completion, characterize it intrinsically if it exists, and establish whether it arises as a regularization-independent limit of nonsingular canonical completions.

Background

The chordal completion theorem guarantees a Lorentz-Gram completion when every fully specified clique principal submatrix is Lorentz-Gram, including singular clique matrices. By contrast, the paper’s canonical clique-product completion, inverse-sparsity characterization, and variational maximum-determinant principle assume that all maximal clique matrices are nonsingular. The unresolved issue is therefore how, or whether, a canonical choice can be made in the singular regime, and whether such a choice is intrinsic rather than dependent on a selected regularization procedure.

References

Is there a distinguished completion for singular admissible data? If so, can it be characterized intrinsically, and is it obtained as a regularization-independent limit of nonsingular canonical completions?

— Hyperbolic distance matrix completion  (2609.10403 - Putinar et al., 9 Sep 2026) in Section ‘Open questions’, Question ‘Canonical completion for singular data’

For a fixed nonchordal graph $G$, however, some admissible partial matrices remain completable. Can these matrices be characterized explicitly? In particular, can global compatibility be detected by conditions associated with the induced cycles of $G$, or with a controlled family of its chordal extensions?

— Hyperbolic distance matrix completion  (2609.10403 - Putinar et al., 9 Sep 2026) in Section ‘Open questions’, Question ‘Completion on a fixed nonchordal graph’