提出新方法恢复含异常值的点集几何结构。
A Dual Basis Approach for Structured Robust Euclidean Distance Geometry
- 利用非正交对偶基构建鲁棒距离几何恢复框架
- 在传感器定位与分子构象数据上表现优于现有方法
- 理论保证可恢复点坐标与相关格拉姆矩阵
欧几里得距离矩阵（EDM）由点集间两两点的平方欧氏距离构成，广泛应用于现代机器学习。本文研究仅通过一组锚点获取其与其他点间距离的情形，此时距离观测存在部分缺失和污染，形成结构性不完整且受干扰的EDM。由于EDM可通过非正交对偶基与半正定格拉姆矩阵关联，受优化领域中非正交对偶基进展启发，本文提出一种新算法框架——基于对偶基的鲁棒欧几里得距离几何恢复（RoDEoDB），用于重构原始点配置。在适度条件下，建立了对格拉姆矩阵与点配置的精确恢复保证。实验表明，RoDEoDB在传感器定位与分子构象数据集上均表现出色。
原文摘要 · Abstract (English)
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.
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