Convergence of the majorization method for multidimensional scaling

Abstract

Multidimensional scaling, Convergence, Step size, Local minima,

Modified starches or stabilizers in preparation of cheese bread

“…As for discrimination of the treatments in relation to the physical-chemical variables for the cheese bread and dough, the multidimensional scaling technique was used, following the procedure recommended by BORG & GROENEN (2005). This reduction was validated considering the approximation between the dissimilarity matrix obtained from the original data and from the variables selected as indicated by DE LEEUW (1988).…”

Graph Drawing by Stress Majorization

“…In this section, we review stress majorization as described in the MDS literature [3,5]. We denote a d-dimensional layout by an n × d matrix X.…”

“…Substantial work in statistical MDS deals with the properties of the majorization process, including proofs of its convergence rate [3]. The MDS literature suggests solving equation (9) by computing (L w ) + , the Moore-Penrose inverse of the singular matrix L w .…”

Multidimensional Scaling by Majorization: A Review

“…Therefore, if the minimum of this quadratic function yields an update with the same ordering (and thus in the same polyhedron), then the update is a local minimum and the next iteration shows an improvement in Stress of zero. This dependency of the update on the rank order of the coordinates was first observed in De Leeuw and Heiser (1977). We can illustrate this effect by applying unidimensional scaling to the cola data by running the code below.…”

“…Initially, the acronym SMACOF stood for 'scaling by majorizing a convex function' following the convex analysis approach of De Leeuw (1977). Here we follow De Heiser (1980) andDe Leeuw (1988) who redefine the acronym as scaling by majorizing a complicated function.…”