Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of -dimensional data items, a configuration of points in -dimensional space is sought that minimizes the so-called stress function . Usually is or , i.e. the matrix lists points in or dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot). The function is a cost or loss function that measures the squared differences between ideal (-dimensional) distances and actual distances in r-dimensional space. It is defined as:
where is a weight for the measurement between a pair of points , is the euclidean distance between and and is the ideal distance between the points (their separation) in the -dimensional data space. Note that can be used to specify a degree of confidence in the similarity between points (e.g. 0 can be specified if there is no information for a particular pair).
A configuration which minimizes gives a plot in which points that are close together correspond to points that are also close together in the original -dimensional data space.
There are many ways that could be minimized. For example, Kruskal[1] recommended an iterative steepest descent approach. However, a significantly better (in terms of guarantees on, and rate of, convergence) method for minimizing stress was introduced by Jan de Leeuw.[2] De Leeuw's iterative majorization method at each step minimizes a simple convex function which both bounds from above and touches the surface of at a point , called the supporting point. In convex analysis such a function is called a majorizing function. This iterative majorization process is also referred to as the SMACOF algorithm ("Scaling by MAjorizing a COmplicated Function").
^Kruskal, J. B. (1964), "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis", Psychometrika, 29 (1): 1–27, doi:10.1007/BF02289565.
^de Leeuw, J. (1977), "Applications of convex analysis to multidimensional scaling", in Barra, J. R.; Brodeau, F.; Romie, G.; et al. (eds.), Recent developments in statistics, pp. 133–145.
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