Doubling the dimension yields a benign landscape for the squared-stress

Abstract: We consider the Euclidean distance geometry problem (EDG): given a subset of the pairwise distances of an unknown cloud of $n$ points in $\mathbb{R}<sup>\ell$, recover the point cloud up to rigid motions. When $n$ is large, a popular practical approach is to minimize a nonconvex quartic, known as the squared-stress or s-stress, over point clouds in $\mathbb{R}<sup>k$, with $k$ potentially larger than $\ell$. It is a long-standing open problem to understand the optimization landscape of the s-stress when all pairwise distances are known (Malone and Trosset, 2000; Parhizkar, 2013). It was recently shown that the landscape is not benign when $k=\ell$, and it was conjectured that the landscape becomes benign as soon as $k\ge \ell+1$ (Song et al., 2025; Criscitiello et al., 2026). Here, we show that the complete-graph s-stress has a benign landscape whenever $k\ge 2(\ell+1)$, establishing the conjecture up to a factor of two. A key idea is to view second-order criticality as a containment of two ellipsoids; finding a descent direction then corresponds to finding a separating hyperplane that violates this containment. This dual perspective yields the stated landscape result, and also applies to any measurement operator whose inverse satisfies a simple frame condition.

• The paper proves that complete-graph squared-stress optimization has a benign landscape for every ground truth and sample size when the optimization dimension satisfies k ≥ 2(ℓ+1), bringing the conjectured threshold k ≥ ℓ+1 within a factor of two.
• The analysis combines a general structured-inverse measurement theorem with dual ellipsoid containment, variational descent directions, and stress-matrix arguments that handle full-rank non-global critical points beyond standard RIP-based methods.
• The codimension-one case establishes benignness at k ≥ ℓ+1 when p−k=1, while incomplete graphs and the full conjectured threshold remain open research problems.

Problem and context

The Euclidean distance geometry (EDG) problem asks to recover a configuration of $n$ points $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$ from a subset of pairwise distances, up to rigid motions. A widely used nonconvex formulation is the squared-stress (s-stress) objective

$$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$

minimized over point clouds in an optimization dimension $k$, possibly larger than the ambient dimension $\ell$. Whether the complete-graph s-stress has a benign landscape—meaning every second-order critical point is globally optimal—is a long-standing question. Prior work established that the landscape is not benign at $k=\ell$ (spurious local minima exist even for $n=\ell+2$ points), and conjectured that benignness holds as soon as $k\ge \ell+1$ (2608.16799).

Main result

The paper proves that benignness holds whenever $k\ge 2(\ell+1)$, for arbitrary ground truths and all $n$, establishing the conjecture up to a factor of two. Because the complete graph is universally rigid, every global minimizer corresponds to the ground truth up to rigid motion even when $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$0. The result is significant algorithmically: local search methods such as gradient descent and trust-region methods provably find global minimizers on benign landscapes, while the factorized formulation optimizes over an $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$1 matrix with $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$2, in contrast to SDP-based methods whose dense $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$3 variables scale quadratically in memory.

The theorem is derived from a more general result for any self-adjoint positive-definite measurement operator $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$4 on a subspace $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$5 of dimension $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$6, whose inverse admits a structured representation: there exist frame atoms $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$7 and $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$8 satisfying a Bessel condition $z_1^\star,\dots,z_n^\star\in\mathbb{R}^\ell$9, norm bounds $$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$0, and $$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$1 where $$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$2. For complete-graph EDG, $$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$3 satisfies these conditions with atoms $$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$4 and $$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$5. The general guarantee is that the landscape is benign if either $$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$6 or

$$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$7

which is implied by $$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$8 since $$s(Z)=\tfrac12\sum_{\{i,j\}\in E}\big(\|z_i-z_j\|^2-d_{ij}^2\big)^2,$$9. Notably, the operator $k$0 has eigenvalues $k$1, $k$2, and $k$3 on centered matrices and fails the restricted isometry property (RIP) used in classical matrix-sensing analyses, so existing RIP-based techniques do not apply.

Two standard reductions structure the argument: second-order critical points are automatically global when $k$4 (the convex regime) or when $k$5 is rank deficient. The core difficulty is therefore full-rank critical points with $k$6. The central intermediate result shows that any full-rank non-global second-order critical point must satisfy

$k$7

where $k$8 spans $k$9; the rank term equals the rank of the least-squares residual $\ell$0, i.e., the dimension of the ground-truth component not explainable by linearly aligning $\ell$1 to $\ell$2.

Proof mechanism: ellipsoid containment and dual probes

A key conceptual contribution is a dual characterization of second-order criticality. Writing $\ell$3 with $\ell$4 orthonormal in $\ell$5, and letting $\ell$6 be the stress matrix, first-order criticality forces $\ell$7 and hence $\ell$8 for some $\ell$9. Second-order criticality is then equivalent to containment of two ellipsoids in $k=\ell$0 ($k=\ell$1):

$k=\ell$2

where $k=\ell$3 encodes the measurement energy $k=\ell$4 over tangent directions and $k=\ell$5. By convex duality, violating containment amounts to finding a separating hyperplane—a "dual probe" $k=\ell$6—whose support function inequality fails. The associated descent direction $k=\ell$7 is defined variationally as the Fenchel conjugate of $k=\ell$8 at $k=\ell$9, and it optimally selects the $n=\ell+2$0-block given the cross-block exposed by $n=\ell+2$1. This refines the descent directions of Criscitiello–McRae–Rebjock–Boumal, which use support points of $n=\ell+2$2 rather than of $n=\ell+2$3; both use the same probes but expose different ellipsoids, and the present choice is sharper.

Under the structured-inverse hypothesis, $n=\ell+2$4 admits an explicit formula via solving a linear system with the matrix $n=\ell+2$5, where $n=\ell+2$6 and $n=\ell+2$7. Positivity of $n=\ell+2$8 follows from its decomposition as a diagonal plus a weighted graph Laplacian. Specializing to rank-one probes $n=\ell+2$9 with $k\ge \ell+1$0 (where $k\ge \ell+1$1), averaging over a covariance $k\ge \ell+1$2, and choosing $k\ge \ell+1$3 yields the intermediate rank bound above. The paper also gives an equivalent randomized proof using Gaussian probes coupling random motions in $k\ge \ell+1$4 with the least-squares residual.

The codimension-one case and evidence at the conjectured threshold

Although computationally unattractive (it requires $k\ge \ell+1$5 for complete-graph EDG), the case $k\ge \ell+1$6 is analyzed in detail because it reaches the conjectured threshold: benignness holds for $k\ge \ell+1$7 when $k\ge \ell+1$8, which covers the first genuinely nonconvex regime beyond the convex one, namely $k\ge \ell+1$9 points with $k\ge 2(\ell+1)$0. At this endpoint, kernel directions alone are insufficient; the vanishing of the generalized Schur complement of $k\ge 2(\ell+1)$1 produces a canonical additional direction—the Schur-companion direction $k\ge 2(\ell+1)$2 orthogonal to $k\ge 2(\ell+1)$3. Combining kernel and companion directions yields a two-dimensional subspace $k\ge 2(\ell+1)$4 on which second-order criticality forces

$k\ge 2(\ell+1)$5

while a purely geometric lemma depending only on the frame atoms shows this quantity is strictly less than $k\ge 2(\ell+1)$6 for every two-dimensional $k\ge 2(\ell+1)$7—a contradiction. The geometric lemma is proved via Ky Fan's principle and Cauchy interlacing together with a simplex inequality, under tight frames, and extended to loose frames by adjoining atoms. The paper suggests that higher-dimensional analogues of Schur-companion directions may be needed to resolve the full conjecture.

Limitations and open questions

Several restrictions are stated plainly. First, all results concern the complete graph; extending to incomplete graphs is left open, and the conjectured threshold $k\ge 2(\ell+1)$8 is known to fail for incomplete graphs already when a single edge is missing. The complete-graph analysis relies crucially on positive semidefiniteness of the stress matrix at first-order critical points, and no comparable positivity property appears to hold for incomplete graphs—even weakly. Second, whether the general condition $k\ge 2(\ell+1)$9 is tight over the class of structured-inverse operators remains open; it is not tight for identity sensing (where benignness already holds at $n$0), and the author does not expect it to be tight for $n$1. Third, the central open question is whether the complete-graph s-stress is benign for all ground truths at $n$2; the codimension-one result provides supporting evidence but not a resolution. Finally, the paper asks whether the descent directions admit a geometric interpretation in terms of point-cloud motions, noting that Procrustes-type residual directions familiar from matrix completion and phase retrieval are insufficient here, and suggesting rigidity theory as a possible source of explanation.

Conclusion

This paper resolves a long-standing landscape question for the complete-graph s-stress up to a factor of two: relaxing the optimization dimension to $n$3 guarantees a benign landscape for arbitrary ground truths, via a general theorem for measurement operators with structured inverses. The dual ellipsoid-containment perspective on second-order criticality, together with variational descent directions and the Schur-companion mechanism in codimension one, constitutes the technical core. The remaining gap between $n$4 and the conjectured $n$5, and the extension to incomplete graphs, remain open.