Benign landscape at the single-dimension relaxation threshold

Establish whether the complete-graph squared-stress objective has a benign landscape for every ground-truth configuration whenever the optimization dimension satisfies k ≥ ℓ + 1.

Background

The paper studies the nonconvex squared-stress objective for Euclidean distance geometry when all pairwise distances are observed. The true configuration lies in dimension ℓ, while optimization is performed in dimension k. The paper proves benignness for k ≥ 2(ℓ + 1), and separately proves the conjectured threshold in the codimension-one regime m = p − k = 1. The general threshold k ≥ ℓ + 1 therefore remains unresolved.

The conjecture is motivated by the existence of spurious local minimizers when k = ℓ and numerical evidence suggesting that a single additional optimization dimension eliminates them. Resolving this question would identify the smallest dimension relaxation guaranteeing the absence of spurious local minima for complete-graph Euclidean distance geometry.

References

On the other hand, based on numerical experiments, \citet{criscitiello2025snl} conjecture that these spurious local minimizers disappear as soon as one relaxes by a single dimension, namely k\ge \ell+1. We establish this conjecture up to a factor of two: we prove that the complete-graph s-stress has no spurious local minima as soon as k\ge 2(\ell+1), regardless of the ground truth.

— Doubling the dimension yields a benign landscape for the squared-stress  (2608.16799 - Criscitiello, 17 Aug 2026) in Section 1, Introduction; Section Perspectives

Is there a geometric interpretation of the descent directions underlying our analysis, at the level of motions of point clouds or their pairwise distances? In many benign-landscape results, the relevant descent directions admit a simple geometric interpretation.

— Doubling the dimension yields a benign landscape for the squared-stress  (2608.16799 - Criscitiello, 17 Aug 2026) in Section Perspectives