Euclidean Distance Matrix (EDM), which consists of pairwise squared Euclidean distances of a given point configuration, finds many applications in modern machine learning. This paper considers the setting where only a set of anchor nodes is used to collect the distances between themselves and the rest. In the presence of potential outliers, it results in a structured partial observation on EDM with partial corruptions. Note that an EDM can be connected to a positive semi-definite Gram matrix via a non-orthogonal dual basis. Inspired by recent development of non-orthogonal dual basis in optimization, we propose a novel algorithmic framework, dubbed Robust Euclidean Distance Geometry via Dual Basis (RoDEoDB), for recovering the Euclidean distance geometry, i.e., the underlying point configuration. The exact recovery guarantees have been established in terms of both the Gram matrix and point configuration, under some mild conditions. Empirical experiments show superior performance of RoDEoDB on sensor localization and molecular conformation datasets.

翻译：欧氏距离矩阵（EDM）由给定点构型的成对欧氏距离平方构成，在现代机器学习中具有广泛应用。本文考虑仅使用一组锚节点来收集其自身与其余节点之间距离的场景。在存在潜在异常值的情况下，这导致对EDM产生具有部分损坏的结构化部分观测。注意到EDM可通过非正交双基与半正定Gram矩阵建立关联。受优化领域中非正交双基最新进展的启发，我们提出了一种新颖的算法框架——基于双基的鲁棒欧氏距离几何方法（RoDEoDB），用于恢复欧氏距离几何（即潜在点构型）。在若干温和条件下，我们针对Gram矩阵和点构型分别建立了精确恢复的理论保证。实证实验表明，RoDEoDB在传感器定位和分子构象数据集上均表现出优越性能。