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 prove among others that PREEEDM converges to a stationary point of PREE, which is also an approximate critical point of REE. Finally, the efficiency of PREEEDM is 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.
Euclidean distance matrix, Euclidean embedding, Majorization and minimization method, Matrix optimization
337–387
Zhou, Shenglong
d183edc9-a9f6-4b07-a140-a82213dbd8c3
Xiu, Naihua
8b5770f7-ae35-4dbe-884a-02fb4ea27bee
Qi, Hou-Duo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
1 September 2020
Zhou, Shenglong
d183edc9-a9f6-4b07-a140-a82213dbd8c3
Xiu, Naihua
8b5770f7-ae35-4dbe-884a-02fb4ea27bee
Qi, Hou-Duo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Zhou, Shenglong, Xiu, Naihua and Qi, Hou-Duo
(2020)
Robust Euclidean embedding via EDM optimization.
Mathematical Programming Computation, 12 (3), .
(doi:10.1007/s12532-019-00168-0).
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 prove among others that PREEEDM converges to a stationary point of PREE, which is also an approximate critical point of REE. Finally, the efficiency of PREEEDM is 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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preeedm_R1
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Zhou2019_Article_RobustEuclideanEmbeddingViaEDM
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Accepted/In Press date: 5 July 2019
e-pub ahead of print date: 9 August 2019
Published date: 1 September 2020
Keywords:
Euclidean distance matrix, Euclidean embedding, Majorization and minimization method, Matrix optimization
Identifiers
Local EPrints ID: 432733
URI: http://eprints.soton.ac.uk/id/eprint/432733
PURE UUID: 473cd5a6-a2a6-4a5b-95cf-69042f8f2dc8
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Date deposited: 25 Jul 2019 16:30
Last modified: 16 Mar 2024 08:02
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Author:
Shenglong Zhou
Author:
Naihua Xiu
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