Localization from structured distance matrices via low-rank matrix recovery

We study the problem of determining the configuration of $n$ points by using their distances to $m$ nodes, referred to as anchor nodes. One sampling scheme is Nystrom sampling, which assumes known distances between the anchors and between the anchors and the $n$ points, while the distances among the $n$ points are unknown. For this scheme, a simple adaptation of the Nystrom method, which is often used for kernel approximation, is a viable technique to estimate the configuration of the anchors and the $n$ points. In this manuscript, we propose a modified version of Nystrom sampling, where the distances from every node to one central node are known, but all other distances are incomplete. In this setting, the standard Nystrom approach is not applicable, necessitating an alternative technique to estimate the configuration of the anchors and the $n$ points. We show that this problem can be framed as the recovery of a low-rank submatrix of a Gram matrix. Using synthetic and real data, we demonstrate that the proposed approach can exactly recover configurations of points given sufficient distance samples. This underscores that, in contrast to methods that rely on global sampling of distance matrices, the task of estimating the configuration of points can be done efficiently via structured sampling with well-chosen reliable anchors. Finally, our main analysis is grounded in a specific centering of the points. With this in mind, we extend previous work in Euclidean distance geometry by providing a general dual basis approach for points centered anywhere.