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A fast matrix majorization-projection method for penalized stress minimization with box constraints

A fast matrix majorization-projection method for penalized stress minimization with box constraints
A fast matrix majorization-projection method for penalized stress minimization with box constraints
Kruskal's stress minimization, though nonconvex and nonsmooth, has been a
major computational model for dissimilarity data in multidimensional scaling.
Semidefinite Programming (SDP) relaxation (by dropping the rank constraint) would lead to a high number of SDP cone constraints.
This has rendered the SDP approach computationally challenging even for problems of small size. In this paper, we reformulate the stress optimization as an
Euclidean Distance Matrix (EDM) optimization with box constraints.
A key element in our approach is the conditional positive semidefinite cone with rank cut.
Although nonconvex, this geometric object allows a fast computation of the projection onto it and it naturally leads to a majorization-minimization algorithm with the minimization step having a closed-form solution. Moreover, we prove that our EDM optimization follows a continuously differentiable path, which greatly facilitated the analysis of the convergence to a stationary point.
The superior performance of the proposed algorithm is demonstrated against some of the state-of-the-art solvers in the field of 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) A fast matrix majorization-projection method for penalized stress minimization with box constraints University of Southampton 16pp. (In Press)

Record type: Monograph (Working Paper)

Abstract

Kruskal's stress minimization, though nonconvex and nonsmooth, has been a
major computational model for dissimilarity data in multidimensional scaling.
Semidefinite Programming (SDP) relaxation (by dropping the rank constraint) would lead to a high number of SDP cone constraints.
This has rendered the SDP approach computationally challenging even for problems of small size. In this paper, we reformulate the stress optimization as an
Euclidean Distance Matrix (EDM) optimization with box constraints.
A key element in our approach is the conditional positive semidefinite cone with rank cut.
Although nonconvex, this geometric object allows a fast computation of the projection onto it and it naturally leads to a majorization-minimization algorithm with the minimization step having a closed-form solution. Moreover, we prove that our EDM optimization follows a continuously differentiable path, which greatly facilitated the analysis of the convergence to a stationary point.
The superior performance of the proposed algorithm is demonstrated against some of the state-of-the-art solvers in the field of sensor network localization and molecular conformation.

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Submitted date: 2017
Accepted/In Press date: 9 June 2018

Identifiers

Local EPrints ID: 417366
URI: https://eprints.soton.ac.uk/id/eprint/417366
PURE UUID: f9940f65-e525-411a-afa3-a04e4a46c664
ORCID for Hou-Duo Qi: ORCID iD orcid.org/0000-0003-3481-4814

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Date deposited: 30 Jan 2018 17:30
Last modified: 14 Mar 2019 01:43

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