Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-Rank Matrix Completion

The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. When the distance information is incomplete, the problem can be formulated as a nuclear norm minimization problem. In this paper, this minimization program is recast as a matrix completion problem of a low-rank
$r$
Gram matrix with respect to a suitable basis. The well-known restricted isometry property cannot be satisfied in this scenario. Instead, a dual basis approach is introduced to theoretically analyze the reconstruction problem. If the Gram matrix satisfies certain coherence conditions with parameter
$\nu $
, the main result shows that the underlying configuration of
$n$
points can be recovered with very high probability from
$O(nr\nu \log ^{2}(n))$
uniformly random samples. Computationally, simple and fast algorithms are designed to solve the Euclidean distance geometry problem. Numerical tests on different 3-D data and protein molecules validate the effectiveness and efficiency of the proposed algorithms.