标题： 黎曼优化在距离几何中的应用：收敛性、鲁棒性和非一致性研究 显示英文标题

标题： Riemannian Optimization for Distance Geometry: A Study of Convergence, Robustness, and Incoherence

摘要： 从部分成对距离中恢复点配置的问题，称为欧几里得距离几何（EDG）问题，在传感器网络定位、分子构象和流形学习等广泛的应用中出现。在本文中，我们提出了一种黎曼优化框架，通过将EDG问题表述为正半定Gram矩阵空间上的低秩矩阵补全任务来解决该问题。可用的距离测量被编码为非正交基中的展开系数，Gram矩阵上的优化通过三角不等式隐式地强制几何一致性，这种结构继承自经典多维缩放。在观察距离的伯努利采样模型下，我们证明了当采样概率满足$p \geq \mathcal{O}(\nu^2 r^2 \log(n)/n)$时，黎曼梯度下降在rank-$r$矩阵流形上以高概率局部线性收敛，其中$\nu$是一个EDG特定的非一致性参数。此外，我们使用一个一步硬阈值过程提供了一个初始化候选，只要采样概率满足$p \geq \mathcal{O}(\nu r^{3/2} \log^{3/4}(n)/n^{1/4})$，就能保证收敛。这项工作的关键技术贡献是对非正交基中对偶基展开产生的对称线性算子的分析，这需要对Hanson--Wright不等式的创新应用，以在耦合项存在的情况下建立最优的限制等距性质。在合成数据上的实验评估表明，我们的算法相对于最先进的方法表现出竞争力。此外，我们提出了一种针对EDG设置的矩阵非一致性的新概念，并为我们的方法提供了鲁棒性保证。 显示英文摘要

摘要： The problem of recovering a configuration of points from partial pairwise distances, referred to as the Euclidean Distance Geometry (EDG) problem, arises in a broad range of applications, including sensor network localization, molecular conformation, and manifold learning. In this paper, we propose a Riemannian optimization framework for solving the EDG problem by formulating it as a low-rank matrix completion task over the space of positive semi-definite Gram matrices. The available distance measurements are encoded as expansion coefficients in a non-orthogonal basis, and optimization over the Gram matrix implicitly enforces geometric consistency through the triangle inequality, a structure inherited from classical multidimensional scaling. Under a Bernoulli sampling model for observed distances, we prove that Riemannian gradient descent on the manifold of rank-$r$ matrices locally converges linearly with high probability when the sampling probability satisfies $p \geq \mathcal{O}(\nu^2 r^2 \log(n)/n)$, where $\nu$ is an EDG-specific incoherence parameter. Furthermore, we provide an initialization candidate using a one-step hard thresholding procedure that yields convergence, provided the sampling probability satisfies $p \geq \mathcal{O}(\nu r^{3/2} \log^{3/4}(n)/n^{1/4})$. A key technical contribution of this work is the analysis of a symmetric linear operator arising from a dual basis expansion in the non-orthogonal basis, which requires a novel application of the Hanson--Wright inequality to establish an optimal restricted isometry property in the presence of coupled terms. Empirical evaluations on synthetic data demonstrate that our algorithm achieves competitive performance relative to state-of-the-art methods. Moreover, we propose a novel notion of matrix incoherence tailored to the EDG setting and provide robustness guarantees for our method.