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The Approximation of Two-Mode Proximity Matrices by Sums of Order-Constrained Matrices

Published online by Cambridge University Press:  01 January 2025

Lawrence Hubert  and
Phipps Arabie
Lawrence Hubert*
Affiliation:
University of Illinois, Champaign Department of Data Theory, University of Leiden
Phipps Arabie
Affiliation:
Faculty of Management, Rutgers University
*
Requests for reprints should be sent to Lawrence Hubert, Department of Psychology, The University of Illinois, 603 East Daniel Street, Champaign, IL 61820.
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Abstract

A least-squares strategy is proposed for representing a two-mode proximity matrix as an approximate sum of a small number of matrices that satisfy certain simple order constraints on their entries. The primary class of constraints considered define Q-forms (or anti-Q-forms) for a two-mode matrix, where after suitable and separate row and column reorderings, the entries within each row and within each column are nondecreasing (or nonincreasing) to a maximum (or minimum) and thereafter nonincreasing (or nondecreasing). Several other types of order constraints are also mentioned to show how alternative structures can be considered using the same computational strategy.

Keywords

two-mode proximity matricesorder constraints(anti-)Robinson form(anti-)Q-formleast-squares matrix approximation
Type
Original Paper
Information
Psychometrika , Volume 60 , Issue 4 , December 1995 , pp. 573 - 605
Copyright
Copyright © 1995 The Psychometric Society

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