AI 中文总结
本文针对欧氏距离几何问题，证明当k≥2(ℓ+1)时，完全图平方应力的优化景观为良性，将k≥ℓ+1的猜想因子缩小一半，采用二阶临界性的对偶视角推导结论。
AI 中文摘要
我们考虑欧氏距离几何问题（EDG）：给定ℝ^ℓ中n个未知点云的部分成对距离，恢复该点云（仅差刚体运动）。当n较大时，常用实用方法是在ℝ^k中的点云上最小化非凸四次函数（称为平方应力或s-stress），其中k可能大于ℓ。当所有成对距离已知时，理解s-stress的优化景观是一个长期未解决的开放问题（Malone和Trosset，2000；Parhizkar，2013）。近期研究表明，当k=ℓ时景观并非良性，且有猜想称当k≥ℓ+1时景观变为良性（Song等，2025；Criscitiello等，2026）。本文证明，当k≥2(ℓ+1)时，完全图s-stress具有良性景观，将该猜想的因子缩小了一半。核心思路是将二阶临界性视为两个椭球的包含关系；寻找下降方向对应于找到违反该包含关系的分离超平面。这种对偶视角得出了上述景观结果，且适用于任何其逆满足简单框架条件的测量算子。
英文摘要
We consider the Euclidean distance geometry problem (EDG): given a subset of the pairwise distances of an unknown cloud of $n$ points in $\mathbb{R}^\ell$, 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 $\mathbb{R}^k$, 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.
Comments38 pages
