Robust Euclidean embedding via EDM optimization

Abstract

This paper aims to propose an efficient numerical method for the most challenging problem known as the robust Euclidean embedding (REE) in the family of multi-dimensional scaling (MDS).The problem is notoriously known to be nonsmooth, nonconvex and its objective is non-Lipschitzian. We first explain that the semidefinite programming (SDP) relaxations and Euclidean distance matrix (EDM) approach, popular for other types of problems in the MDS family, failed to provide a viable method for this problem.We then propose a penalized REE (PREE), which can be economically majorized. We show that the majorized problem is convex provided that the penalty parameter is above certain threshold. Moreover, it has a closed-form solution, resulting in an efficient algorithm dubbed as \PREEEDM\ (for Penalized REE via EDM optimization).We will prove among others that \PREEEDM\ converges to a stationary point of PREE. Finally, the efficiency of \PREEEDM\ will be compared with several state-of-the-art methods including SDP and EDM solvers on a large number of test problems from sensor network localization and molecular conformation.

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ORCID for Hou-Duo Qi: orcid.org/0000-0003-3481-4814

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