Localization From Incomplete Euclidean Distance Matrix: Performance Analysis for the SVD–MDS Approach

Localizing a cloud of points from noisy measurements of a subset of pairwise distances has applications in various areas, such as sensor network localization and reconstruction of protein conformations from nuclear magnetic resonance measurements. Drineas et al. proposed a natural two-stage approach, named singular value decomposition (SVD)–multidimensional scaling (MDS), for this purpose at the 2006 3rd Annual IEEE Communications Society on Sensor and Ad Hoc Communications and Networks. This approach consists of a low-rank matrix completion algorithm, named SVD-reconstruct, to estimate random missing distances, and the classic MDS method to estimate the positions of nodes. In this paper, we present a detailed analysis for this method. More specifically, we first establish error bounds for Euclidean distance matrix completion in both expectation and tail forms. Utilizing these results, we then derive the error bound for the recovered positions of nodes. In order to assess the performance of SVD-reconstruct, we present the minimax lower bound of the zero-diagonal, symmetric, low-rank matrix completion problem by Fano's method. This result reveals that when the noise level is low, the SVD-reconstruct approach for Euclidean distance matrix completion is suboptimal in the minimax sense; when the noise level is high, SVD-reconstruct can achieve the optimal rate up to a constant factor.