eflecting the importance or precision of dissimilarity # i j . 1. SOURCES OF DISTANCE DATA Dissimilarity information about a set of objects can arise in many different ways. We review some of the more important ones, organized by scientific discipline. 1.1. Geodesy. The most obvious application, perhaps, is in sciences in which distance is measured directly, although generally with error. This happens, for instance, in triangulation in geodesy. We have measurements which are approximately equal to distances, either Euclidean or spherical, depending on the scale of the experiment. In other examples, measured distances are less directly related to physical distances. For example, we could measure airplane or road or train travel distances between different cities. Physical distance is usually not the only factor determining these types of dissimilarities. 1 2 J. DE LEEUW<

This dissertation examines the use of adaptive methods to automatically improve the performance of ranked text retrieval systems. The goal of a ranked retrieval system is to manage a large collection of text documents and to order documents for a user based on the estimated relevance of the documents to the user's information need (or query). The ordering enables the user to quickly find documents of interest. Ranked retrieval is a difficult problem because of the ambiguity of natural language, the large size of the collections, and because of the varying needs of users and varying collection characteristics. We propose and empirically validate general adaptive methods which improve the ability of a large class of retrieval systems to rank documents effectively. Our main adaptive method is to numerically optimize free parameters in a retrieval system by minimizing a non-metric criterion function. The criterion measures how well the system is ranking documents relative to a target ordering, defined by a set of training queries which include the users' desired document orderings. Thus, the system learns parameter settings which better enable it to rank relevant documents before irrelevant. The non-metric approach is interesting because it is a general adaptive method, an alternative to supervised methods for training neural networks in domains in which rank order or prioritization is important. A second adaptive method is also examined, which is applicable to a restricted class of retrieval systems but which permits an analytic solution. The adaptive methods are applied to a number of problems in text retrieval to validate their utility and practical efficiency. The applications include: A dimensionality reduction of vector-based document representations to a vector spa...

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This paper considers numerical algorithms for finding local minimizers of metric multidimensional scaling problems. Both the STRESS and SSTRESS criteria are considered, and the leading algorithms for each are carefully explicated. A new algorithm, based on Newton's method, is proposed. Translational and rotational indeterminancy is removed by a parametrization that has not previously been used in multidimensional scaling algorithms. In contrast to previous algorithms, a very pleasant feature of the new algorithm is that it can be used with either the STRESS or the SSTRESS criterion. Numerical results are presented. Key words: Metric multidimensional scaling, STRESS criterion, SSTRESS criterion, unconstrained optimization, Newton's method. Department of Computational and Applied Mathematics, Rice University, Houston, TX 77251-1892. This author was generously supported by a Patricia R. Harris Fellowship. y Department of Computational and Applied Mathematics and Center for Research in...

Least squares multidimensional scaling is known to have a serious problem of local minima, especially if one dimension is chosen, or if city-block distances are involved. One particular strategy, the smoothing strategy proposed by Pliner (1986, 1996), turns out to be quite successful in these cases. Here, we propose a slightly different approach, called distance smoothing. We extend distance smoothing for any Minkowski distance and show that the S-Stress loss function is a special case. In addition, we extend the majorization approach to multidimensional scaling to have a one-step update for Minkowski parameters larger than 2 and use the results for distance smoothing. We present simple ideas for finding quadratic majorizing functions. The performance of distance smoothing is investigated in several examples, including two simulation studies.

this paper we review the problem of fitting Euclidean distances to data, using a least squares loss function. This problem must be distinguished from the problem of least squares fitting squared Euclidean distances to data, or the problem of least squares fitting scalar products to data. A comparison of these three different approaches to multidimensional scaling is given by De Leeuw and Heiser [9] and Meulman [23]. Recent developments in squared distance scaling are reviewed in a nice paper by Glunt e.a. [13]. We shall prove some existence-type results and some convergence type results. They are both derived by using a constructive proof method, the majorization method . The method was introduced in 1977 in MDS by De Leeuw [4]. A simplified treatment of the Euclidean case was published in 1978, with some extensions to individual differences scaling [7]. Individual differences scaling was subsequently recognized to be a special scaling problem with restrictions on the configuration, and a general approach to such restricted MDS problems was presented in [8]. Speed of convergence of the algorithm was studied in [6]. Other reviews, with some additional extensions, are [16] and [22]. Although [8] did use the majorization method to prove necessary conditions for an extremum, this has not really been followed up in the scaling and multivariate analysis literature. In this paper we start with the basic results for the metric case, and then introduce various generalizations. Most of what we discuss is a review of results that have been published elsewhere, but some of it is new. I think the paper illustrates the remarkable power of the majorization method. 1: THE LOSS FUNCTION

Multidimensional scaling (MDS) is a collection of data analytic techniques for constructing configurations of points from dissimilarity information about interpoint distances. A popular measure of the fit of the constructed distances to the observed dissimilarities is the stress criterion, which must be minimized by numerical optimization. Empirical evidence concerning the existence of nonglobal minimizers of the stress criterion is somewhat contradictory. We report a configuration that we have demonstrated to be a nonglobal minimizer. 1 Preliminaries Multidimensional scaling (MDS) is a collection of techniques for fitting distance models to distance data. The data are called dissimilarities. Formally, a symmetric n \Theta n matrix \Delta = (ffi ij ) is a dissimilarity matrix if ffi ij 0 and ffi ii = 0. In this report, we restrict attention to the case of a single dissimilarity matrix (two-way MDS). The goal of MDS is to construct a configuration of points in a target metric (usually...

Constrained multidimensional scaling was put on a firm theoretical basis by Jan De Leeuw and Willem Heiser in the 1980's. There is a simple method of fitting, based on distance via innerproducts, and a numerically more complicated one that is truly based on least-squares on distances. The unconstrained forms are known as principal coordinate analysis and nonmetric multidimensional scaling, respectively. Constraining the solution by external variables brings the power of classical regression analysis back into multidimensional data analysis. This idea is developed and illustrated, with emphasis on constrained principal coordinate analysis.

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