Sensitivity analysis of the strain criterion for multidimensional scaling

Authors:
R. M. Lewis;M. W. Trosset
Affiliations:
Department of Mathematics, College of William & Mary, P.O. Box 8795, Williamsburg, VA 23185-8795, USA;Department of Mathematics, College of William & Mary, P.O. Box 8795, Williamsburg, VA 23185-8795, USA
Venue:
Computational Statistics & Data Analysis
Year:
2006

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Abstract

Multidimensional scaling (MDS) is a collection of data analytic techniques for constructing configurations of points from dissimilarity information about interpoint distances. Classsical MDS assumes a fixed matrix of dissimilarities. However, in some applications, e.g., the problem of inferring 3-dimensional molecular structure from bounds on interatomic distances, the dissimilarities are free to vary, resulting in optimization problems with a spectral objective function. A perturbation analysis is used to compute first- and second-order directional derivatives of this function. The gradient and Hessian are then inferred as representers of the derivatives. This coordinate-free approach reveals the matrix structure of the objective and facilitates writing customized optimization software. Also analyzed is the spectrum of the Hessian of the objective.