Localization from structured distance matrices via low-rank matrix recovery

TL;DR

The paper tackles localizing $n$ points from partial distance information to $m$ anchors by introducing a modified Nyström sampling model that uses a central anchor with complete row distances, enabling anchorless localization. It reframes the problem as recovering a low-rank submatrix $[\mathbf{A}\ \mathbf{B}]$ of the Gram matrix $\mathbf{K}$ via a dual-basis approach, and provides two convex optimization programs to recover the needed blocks, followed by Nyström to obtain the full configuration. The authors extend the centering to arbitrary barycentric weights and demonstrate exact or near-exact recovery on synthetic and protein datasets, highlighting the practical viability of structured distance sampling over global, fully observed distance matrices. This work advances Euclidean distance geometry and localization by enabling efficient, anchor-aware and anchorless recovery under incomplete, structured distance measurements with theoretical and empirical support.

Abstract

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.

Figures (5)

• Figure 2: Dual basis objects $\bm{w}_{1,2}, \bm{v}_{1,2}$ for $n=4$.
• Figure 3: Illustration of the sampling model with complete anchor-anchor distances and partial observations for anchor-mobile distances. The distances between the mobile nodes are not sampled. In addition, we assume that we know the distance of all mobile nodes from one of the anchors.
• Figure 4: Recovered points $\bm{P}$ from the same distances $\{3, 4, 5\}$, but with different choices of $\bm{s}$: $(1/3, 1/3, 1/3)$, $(1, 0, 0)$, and $(-3, 4, 0)$, respectively. The squared distances of recovered points from the origin are given by $-\frac{1}{2}(\bm{s}^{\top}\bm{D}\bm{s})\bm{1} + \bm{D}\bm{s}$gower1985properties.
• Figure 5: Target structure 1PTQ (in green) and numerically estimated structure (in red). A realization from the experiment with $\gamma = 0.2$ and $\alpha=6$ (RMSE = 0.65).
• Figure 6: Target structure 1AX8 (in green) and numerically estimated structure (in red). A realization from the experiment with $\gamma = 0.2$ and $\alpha=9$ (RMSE = 0.17).

Theorems & Definitions (16)

• Theorem 1
• proof
• Corollary 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof