The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) 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 EDMC 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 nonnegativity and 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 $p\geq O(ν^2 r^2\log(n)/n)$, where $ν$ is an EDMC-specific incoherence parameter. Furthermore, we provide an initialization candidate using a one-step hard thresholding procedure that yields convergence, provided the sampling probability satisfies $p \geq O(νr^{3/2}\log^{3/4}(n)/n^{1/4})$. 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 analysis of a second order degenerate U-statistic 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 provide a geometric interpretation of matrix incoherence tailored to the EDMC setting and provide robustness guarantees for our method.

翻译：从部分成对距离中恢复点配置的问题，即欧氏距离矩阵完备化问题，在传感器网络定位、分子构象和流形学习等多个领域有广泛应用。本文提出一种黎曼优化框架，通过将问题表述为半正定Gram矩阵空间上的低秩矩阵完备化任务来解决EDMC问题。可用的距离测量被编码为非正交基中的展开系数，在Gram矩阵上进行优化通过非负性和三角不等式隐式地强制几何一致性，这一结构继承自经典的多维缩放。在观测距离的伯努利采样模型下，我们证明当采样概率满足$p\geq O(ν^2 r^2\log(n)/n$时（其中$ν$是EDMC特定的不相干参数），秩为$r$的流形上的黎曼梯度下降以高概率局部线性收敛。此外，我们提供一种基于一步硬阈值过程的初始化候选方案，在采样概率满足$p \geq O(νr^{3/2}\log^{3/4}(n)/n^{1/4})$时保证收敛。本文的关键技术贡献在于分析了非正交基中对偶基展开产生的对称线性算子，这需要分析二阶退化U型统计量以在存在耦合项的情况下建立最优受限等距性质。在合成数据上的实验评估表明，我们的算法相对于现有最优方法取得了具有竞争力的性能。此外，我们为EDMC场景提供了矩阵不相干的几何解释，并为方法提供了鲁棒性保证。