We study the problem of determining the configuration of $n$ points, referred to as mobile nodes, by utilizing pairwise distances to $m$ fixed points known as anchor nodes. In the standard setting, we have information about the distances between anchors (anchor-anchor) and between anchors and mobile nodes (anchor-mobile), but the distances between mobile nodes (mobile-mobile) are not known. For this setup, the Nystr\"om method is a viable technique for estimating the positions of the mobile nodes. This study focuses on the setting where the anchor-mobile block of the distance matrix contains only partial distance information. First, we establish a relationship between the columns of the anchor-mobile block in the distance matrix and the columns of the corresponding block in the Gram matrix via a graph Laplacian. Exploiting this connection, we introduce a novel sampling model that frames the position estimation problem as low-rank recovery of an inner product matrix, given a subset of its expansion coefficients in a special non-orthogonal basis. This basis and its dual basis--the central elements of our model--are explicitly derived. Our analysis is grounded in a specific centering of the points that is unique to the Nystr\"om method. With this in mind, we extend previous work in Euclidean distance geometry by providing a general dual basis approach for points centered anywhere.

翻译：摘要：本文研究利用$n$个点（称为移动节点）与$m$个固定点（称为锚节点）之间的成对距离，来确定移动节点配置的问题。在标准设定中，我们拥有锚节点之间（锚-锚）以及锚节点与移动节点之间（锚-移动）的距离信息，但移动节点之间（移动-移动）的距离未知。针对这一设定，Nyström方法成为估计移动节点位置的一种可行技术。本研究聚焦于距离矩阵中锚-移动块仅包含部分距离信息的情形。首先，我们通过图拉普拉斯算子，建立了距离矩阵中锚-移动块列与格拉姆矩阵对应块列之间的关系。利用这一联系，我们提出了一种新颖的采样模型，将位置估计问题转化为在给定内积矩阵在特定非正交基上展开系数子集的条件下，对该矩阵的低秩恢复问题。该基及其对偶基（我们模型的核心元素）被显式推导出来。我们的分析基于一种特定的、仅适用于Nyström方法的点中心化处理。基于此，我们通过提供一种针对任意中心化点的通用对偶基方法，扩展了欧几里得距离几何领域的先前工作。