General amplification for signed optimization landscapes

PDF p. 39: Section 7, paragraph on Burer–Monteiro landscapes

Can the expansion and amplification techniques used for synchronization be generalized to control optimization landscapes arising in statistical inference when the adjacency matrix has signed entries? In particular, can such techniques address Burer–Monteiro relaxations for community detection, and can the newer line of work described by the authors recover a result as general as their Theorem 1.10?

For a concrete formulation of the signed landscape, let $B$ be a real symmetric $n\times n$ matrix, possibly with negative entries, and let $Y\in {\mathbb{R}}^{n\times r}$ have unit-length rows $u_i$. One considers the objective $F_B(Y)=\mathrm{Tr}(Y^{T}BY)={\sum }_{i,j}{B}_{ij}{u}_i\cdot u_j$ on these configurations. The matrix $X=Y{Y}^T$ is their Gram matrix, with $X_{ij}=u_i\cdot u_j$; using such a factor to parametrize a positive-semidefinite matrix with diagonal entries one is the Burer–Monteiro factorization, with rank at most $r$. A landscape describes the objective’s local extrema and critical points on the feasible configurations. Community detection is the inference of groups of vertices from graph data; the exact statistical model, desired landscape conclusion, rank and signed spectral hypotheses must be specified in any proposed theorem.

The comparison theorem gives a sufficient two-sided spectral expansion criterion for global synchronization of an unsigned graph. Global synchronization means that, outside a phase-measurable set of phase-volume zero, the trajectory from every initial phase configuration converges to a common phase in the rotating frame of the homogeneous Kuramoto system; the limiting common phase may depend on the initial configuration. The local supplier states the criterion under countable choice and positive reference degree. A signed analogue is not asserted by that theorem.

Signed vector-spin objectives and low-rank Gram factors

General two-sided spectral criterion — Theorem 1.10

The Axiom of Countable Choice (AC_ω)

Globally synchronizing graph — Definition 1.1

The question remains unresolved in this item as of 2026-11-14; no solution or exhaustive current literature assessment is claimed. On PDF page 39, the article identifies its repeated indirect use of nonnegative adjacency entries as an obstacle. It reports higher-rank progress by Abdalla, Bandeira, Boumal and McRae using different optimization tools; it also describes rank reduction by Rakoto Endor and Waldspurger and subsequent work by McRae. The article says it is unclear whether that newer line recovers a result as general as Theorem 1.10.

The abstract of McRae, Abdalla, Bandeira and Boumal reports model-specific asymptotically benign landscapes near exact-recovery thresholds when the finite rank is sufficiently large. This is contextual partial progress, not a solution of the general question stated above. No proof of that result was examined here.

The question and the article’s description of the obstacle and subsequent progress are attributed to Expander graphs are globally synchronizing, PDF page 39, Section 7. The contextual partial result is described in the abstract of Nonconvex landscapes for Z₂ synchronization and graph clustering are benign near exact recovery thresholds. Neither citation proves a general signed analogue.