Euclidean Distance Reconstruction from Partial Distance Information

Euclidean distance matrix completion problem aims at reconstructing the low dimensional geometry structure of nodes given only a few pairwise distances between nodes [1]. This problem arises in many applications including networks and machine learning where much information of points is laking. For instance, in sensor networks, due to the constraints of energy and communication radius, sensor nodes can only measure the pairwise distances accurately between their nearby nodes. As a consequence, we can only form a partial Euclidean distance matrix. Now, an interesting question is: given this partial Euclidean distance matrix, can we reconstruct or estimate the Euclidean distance matrix exactly? Many works are based on heuristic algorithms without rigorous mathematical justification. However, recent landmark work in low-rank matrix reconstruction [2][3][4] provides some new directions on attacking this problem. One of the central message of these works is the feasibility of exact matrix reconstruction by solving a tractable semidefinite programming problem, with a few entries at hand. We notice that the Euclidean distance matrix is extremely low rank considering the fact that nodes are located in low-dimensional space (typically of dimensions two or three). In particular, the rank of the Euclidean distance matrix is at most d + 2, where d is the dimension of the coordinates of node locations. This fact motivates us to apply the low rank matrix completion theory to the Euclidean distance matrix completion problem.