Euclidean Distance Matrix Correction with a Single Corrupted Element

Abstract

In this thesis, we study the problem of correcting the error from a noisy Euclidean distance matrix (EDM). An EDM is a matrix where elements are squared distance between points in R^d. We consider the special case where only one of the distance is corrupted.

We develop efficient algorithms to solve this problem, initially assuming that the points are in general position, and solve the problem using three different types of facial reduction: exposing vector, facial vectors, and Gale transform. Furthermore, we investigate yielding elements of an EDM and develop an algorithm for identifying one small principal submatrix with the embedding dimension d when many of the points are in a linear manifold of dimension smaller than d, allowing us to handle a more general problem. We present numerical experiments implemented in MATLAB, demonstrating the effectiveness of our solutions.