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

Distance geometry problems (EDG) are central to fields like molecular conformation and sensor network localization. Classical methods such as multidimensional scaling (MDS) and semidefinite programming (SDP) have provided solutions, but scalability remains a challenge. Recent advances in non-convex formulations, especially squared-stress minimization, have shown empirical success, yet theoretical guarantees are limited. Prior work identified the existence of spurious local minima when the embedding dimension equals the true dimension, with conjectures suggesting that increasing the dimension beyond ℓ+1 could eliminate these traps. However, rigorous proofs were lacking, and the landscape's structure in high dimensions was poorly understood. This paper builds on these foundations, aiming to establish precise conditions under which the non-convex landscape becomes benign, thus enabling reliable and efficient algorithms.

The core challenge is to characterize the optimization landscape of the squared-stress function in high-dimensional settings, particularly when the embedding dimension k exceeds the true dimension ℓ. The key question is whether increasing k can guarantee the absence of non-global local minima, facilitating convergence of local search algorithms. Existing results are either limited to the minimal dimension (k=ℓ) with known traps or require overly conservative bounds. The difficulty lies in analyzing second-order critical points, which may correspond to saddle points or spurious minima, especially under the complex geometry of the measurement space. Overcoming this requires innovative geometric and spectral techniques to understand the structure of critical points and their global optimality.

The main innovation is the geometric reinterpretation of second-order critical points as an ellipsoid containment problem, enabling a dual geometric analysis. This approach leverages the spectral properties of measurement operators satisfying simple frame conditions, extending the benign landscape guarantee from k≥ℓ to k≥2(ℓ+1). The introduction of Schur-companion directions enriches the analysis, allowing the authors to handle special cases where n≤ℓ+3. The framework generalizes previous results, providing a unified geometric perspective that can be applied to a broad class of measurement operators, not just Euclidean distance matrices. This significantly advances the theoretical understanding of non-convex optimization landscapes in distance geometry.

• �� Define the squared-stress objective and its geometric properties. • Reformulate second-order criticality as an ellipsoid containment problem using duality. • Analyze the spectral properties of measurement operators satisfying frame conditions. • Derive conditions under which all second-order critical points are globally optimal for k≥2(ℓ+1). • Introduce Schur-companion directions to handle special cases, such as n≤ℓ+3. • Extend the analysis to measurement operators with simple inverse frame conditions, broadening applicability. • Use matrix spectral analysis, geometric duality, and residual directions to establish the main theorem.

The paper primarily provides theoretical proofs, supported by numerical simulations. Simulations involve generating random point clouds in ℝ^ℓ, constructing complete graphs, and solving the squared-stress minimization via gradient-based algorithms. Results show that for k≥2(ℓ+1), the algorithms consistently converge to the global minimum, with no evidence of spurious local minima. Additional experiments test the special case k=ℓ+1 with n≤ℓ+3, confirming the benign landscape. Noise robustness and incomplete graph scenarios are also preliminarily explored, indicating the theoretical bounds hold under mild perturbations and partial measurements.

Simulations confirm that when k≥2(ℓ+1), optimization algorithms reliably reach the global minimum, matching theoretical predictions. For k=ℓ+1 and n≤ℓ+3, the landscape remains benign, validating the near-optimal threshold. The geometric analysis demonstrates that all second-order critical points are globally optimal under these conditions, effectively eliminating traps. The results also suggest that the measurement operator's frame condition is crucial for the theoretical guarantees, with potential extensions to broader classes of operators.

The findings directly impact large-scale point cloud reconstruction, molecular conformation modeling, and sensor network localization. By relaxing the dimension requirement, algorithms can operate efficiently in higher dimensions without risking local minima traps. The theoretical guarantees facilitate the design of robust, scalable algorithms for real-world applications where measurements are complete or nearly complete, and noise levels are manageable. This work also informs the development of new non-convex optimization techniques that leverage geometric insights for practical problems in engineering and data science.

Current results are primarily for complete graphs; real-world scenarios often involve incomplete or noisy measurements. Extending the guarantees to these cases remains an open challenge. The dimension condition k≥2(ℓ+1), while significant, still imposes a higher computational burden compared to the minimal ℓ. The analysis relies on specific frame conditions, which may not hold in all measurement models. Future work should focus on relaxing these assumptions, improving robustness, and developing efficient algorithms for large-scale, noisy data.