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Robust Euclidean embedding via EDM optimization

Robust Euclidean embedding via EDM optimization
Robust Euclidean embedding via EDM optimization
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.
University of Southampton
Zhou, Shenglong
4d6bd93c-6940-4177-9874-e0349214d4c2
Xiu, Naihua
8b5770f7-ae35-4dbe-884a-02fb4ea27bee
Qi, Hou-Duo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Zhou, Shenglong
4d6bd93c-6940-4177-9874-e0349214d4c2
Xiu, Naihua
8b5770f7-ae35-4dbe-884a-02fb4ea27bee
Qi, Hou-Duo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85

Zhou, Shenglong, Xiu, Naihua and Qi, Hou-Duo (2018) Robust Euclidean embedding via EDM optimization University of Southampton 43pp.

Record type: Monograph (Working Paper)

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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Published date: 20 March 2018

Identifiers

Local EPrints ID: 420991
URI: https://eprints.soton.ac.uk/id/eprint/420991
PURE UUID: 75346a2a-70ae-47fe-9fa4-666e743107f8
ORCID for Hou-Duo Qi: ORCID iD orcid.org/0000-0003-3481-4814

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Date deposited: 21 May 2018 16:30
Last modified: 14 Mar 2019 01:43

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