On the Use of Second-Order Information in
Distance Geometry and Multidimensional Scaling
Michael Trosset
Department of Mathematics
College of William and Mary
Tuesday, June 4, 2002 15:00-16:00,
Room 145, NIST North (820)
Gaithersburg
Tuesday, June 4, 2002 13:00-14:00,
Room 4550
Boulder
Abstract:
Multidimensional Scaling (MDS) is a collection of techniques for
constructing configurations of points from information about
interpoint distances. Originally developed for psychometric
applications, MDS is now widely used to visualize multivariate
data sets. Important contemporary applications include the
problem of inferring the 3-dimensional structure of a molecule
from information about its interatomic distances.
MDS can be formulated as a collection of optimization problems,
most of which require numerical solution. Most well-known MDS
algorithms are actually first-order (gradient) methods for
solving specific optimization problems. Some researchers have
argued that second-order methods are inappropriate for MDS.
This talk presents the case for second-order methods, arguing
that the standard objective functions used to formulate MDS have
low curvature near solutions and that second-order information is
essential if one hopes to compute accurate solutions. I will
describe how the use of second-order information led to new
insights about the prevalence of nonglobal minimizers of the raw
stress criterion, summarize Sibson's (1979) perturbational
analysis of classical MDS, and discuss my own efforts to extend
classical MDS from the case of fixed dissimilarities to the case
of bound-constrained dissimilarity variables.
Contact: A. J. KearsleyNote: Visitors from outside NIST must contact
Robin Bickel; (301) 975-3668;
at least 24 hours in advance.
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