A novel low-rank matrix completion approach to estimate missing entries in Euclidean distance matrices

TL;DR

This paper introduces a low-rank matrix completion method leveraging the properties of Euclidean Distance Matrices to accurately estimate missing distances, demonstrating high precision and efficiency in large-scale datasets.

Contribution

It presents a novel SVD-based approach that exploits the low-rank structure of EDMs, improving accuracy and convergence speed over existing methods.

Findings

Successfully recovered large EDMs with up to 98% missing data

Achieved high precision in estimating missing distances

Required fewer iterations than state-of-the-art techniques

Abstract

A Euclidean Distance Matrix (EDM) is a table of distance-square between points on a k- dimensional Euclidean space, with applications in many fields (e.g. engineering, geodesy, economics, genetics, biochemistry, psychology). A problem that often arises is the absence (or uncertainty) of some EDM elements. In many situations, only a subset of all pairwise distances is available and it is desired to have some procedure to estimate the missing distances. In this paper, we address the problem of missing data in EDM through low-rank matrix completion techniques. We exploit the fact that the rank of a EDM is at most k+2 and does not depend on the number of points, which is, in general, much bigger then k. We use a Singular Value Decomposition approach that considers the rank of the matrix to be completed and computes, in each iteration, a parameter that controls the convergence of the method. After performing a number of computational experiments, we could observe that our proposal was able to recover, with high precision, random EDMs with more than one thousand points and up to 98 percent of missing data in few minutes. Additionally, our method required a smaller number of iterations when compared to other competitive state-of-art technique.