NDLI: Constrained Best Euclidean Distance Embedding on a Sphere: A Matrix Optimization Approach

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Xiu, Naihua Bai, Shuanghua Qi, Huo-Duo
Copyright Year	2015
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 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.
Starting Page	439
Ending Page	467
Page Count	29
File Format	PDF
ISSN	10526234
DOI	10.1137/13094918X
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	1
Volume Number	25
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-02-26
Access Restriction	Subscribed
Subject Keyword	Lagrangian duality matrix optimization semismooth Newton-CG method spherical multidimensional scaling Euclidean distance matrix Complementarity and equilibrium problems and variational inequalities Convex programming
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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