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Constrained best Euclidean distance embedding on a sphere: a matrix optimization approach

Constrained best Euclidean distance embedding on a sphere: a matrix optimization approach
Constrained best Euclidean distance embedding on a sphere: a matrix optimization approach
The problem of data representation on a sphere of unknown radius arises from various disciplines such as Statistics (spatial data representation), Psychology (constrained multidimensional scaling), and Computer Science (machine learning and pattern recognition). The best representation often needs to minimize a distance function of the data on a sphere as well as to satisfy some Euclidean distance constraints. It is those spherical and Euclidean distance constraints that present an enormous challenge to the existing algorithms. In this paper, we reformulate the problem as an Euclidean distance matrix optimization problem with a low rank constraint. We then propose an iterative algorithm that uses a quadratically convergent Newton-CG method at its each step. We study fundamental issues including constraint nondegeneracy and the nonsingularity of generalized Jacobian that ensure the quadratic convergence of the Newton method. We use some classic examples from the spherical multidimensional scaling to demonstrate the flexibility of the algorithm in incorporating various constraints. We also present an interesting application to the circle fitting problem
1052-6234
439-467
Bai, Shuanghua
aa4f4cf4-7032-4c79-9214-79822dc29d27
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Xiu, Naihua
8e84e128-101b-4b57-aa47-e6002470ae9d
Bai, Shuanghua
aa4f4cf4-7032-4c79-9214-79822dc29d27
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Xiu, Naihua
8e84e128-101b-4b57-aa47-e6002470ae9d

Bai, Shuanghua, Qi, Houduo and Xiu, Naihua (2015) Constrained best Euclidean distance embedding on a sphere: a matrix optimization approach. SIAM Journal on Optimization, 25 (1), 439-467. (doi:10.1137/13094918X).

Record type: Article

Abstract

The problem of data representation on a sphere of unknown radius arises from various disciplines such as Statistics (spatial data representation), Psychology (constrained multidimensional scaling), and Computer Science (machine learning and pattern recognition). The best representation often needs to minimize a distance function of the data on a sphere as well as to satisfy some Euclidean distance constraints. It is those spherical and Euclidean distance constraints that present an enormous challenge to the existing algorithms. In this paper, we reformulate the problem as an Euclidean distance matrix optimization problem with a low rank constraint. We then propose an iterative algorithm that uses a quadratically convergent Newton-CG method at its each step. We study fundamental issues including constraint nondegeneracy and the nonsingularity of generalized Jacobian that ensure the quadratic convergence of the Newton method. We use some classic examples from the spherical multidimensional scaling to demonstrate the flexibility of the algorithm in incorporating various constraints. We also present an interesting application to the circle fitting problem

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Accepted/In Press date: 1 December 2014
e-pub ahead of print date: 26 February 2015
Published date: 2015
Organisations: Operational Research

Identifiers

Local EPrints ID: 372356
URI: http://eprints.soton.ac.uk/id/eprint/372356
ISSN: 1052-6234
PURE UUID: aedc1e2e-1305-43db-b0e0-726f670e51d1
ORCID for Houduo Qi: ORCID iD orcid.org/0000-0003-3481-4814

Catalogue record

Date deposited: 08 Dec 2014 14:20
Last modified: 24 Jun 2020 00:30

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