Stress majorization

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Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of n m-dimensional data items, a configuration X of n points in r(<<m)-dimensional space is sought that minimizes the so called stress function \sigma(X). Usually r is 2 or 3, i.e. the (r x n) matrix X lists points in 2- or 3-dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot). The function \sigma is a cost or loss function that measures the squared differences between ideal (m-dimensional) distances and actual distances in r-dimensional space. It is defined as:

\sigma(X)=\sum_{i<j\le n}w_{ij}(d_{ij}(X)-\delta_{ij})^2

where w_{ij}\ge 0 is a weight for the measurement between a pair of points (i,j), d_{ij}(X) is the euclidean distance between i and j and \delta_{ij} is the ideal distance between the points (their separation) in the m-dimensional data space. Note that w_{ij} 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 X which minimizes \sigma(X) gives a plot in which points that are close together correspond to points that are also close together in the original m-dimensional data space.

There are many ways that  \sigma(X) 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 \sigma from above and touches the surface of \sigma at a point Z, 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 convex function").

The SMACOF algorithm[edit]

The stress function \sigma can be expanded as follows:


\sigma(X)=\sum_{i<j\le n}w_{ij}(d_{ij}(X)-\delta_{ij})^2
=\sum_{i<j}w_{ij}\delta_{ij}^2 + \sum_{i<j}w_{ij}d_{ij}^2(X)-2\sum_{i<j}w_{ij}\delta_{ij}d_{ij}(X)

Note that the first term is a constant C and the second term is quadratic in X (i.e. for the Hessian matrix V the second term is equivalent to trX'VX) and therefore relatively easily solved. The third term is bounded by:


\sum_{i<j}w_{ij}\delta_{ij}d_{ij}(X)=\,\operatorname{tr}\, X'B(X)X \ge \,\operatorname{tr}\, X'B(Z)Z

where B(Z) has:

b_{ij}=-\frac{w_{ij}\delta_{ij}}{d_{ij}(Z)} for d_{ij}(Z)\ne 0, i \ne j

and b_{ij}=0 for d_{ij}(Z)=0, i\ne j

and b_{ii}=-\sum_{j=1,j\ne i}^n b_{ij}.

Proof of this inequality is by the Cauchy-Schwarz inequality, see Borg[3] (pp. 152–153).

Thus, we have a simple quadratic function \tau(X,Z) that majorizes stress:

\sigma(X)=C+\,\operatorname{tr}\, X'VX - 2 \,\operatorname{tr}\, X'B(X)X
\le C+\,\operatorname{tr}\, X' V X - 2 \,\operatorname{tr}\, X'B(Z)Z = \tau(X,Z)


The iterative minimization procedure is then:

  • at the kth step we set Z\leftarrow X^{k-1}
  • X^k\leftarrow \min_X \tau(X,Z)
  • stop if \sigma(X^{k-1})-\sigma(X^{k})<\epsilon otherwise repeat.

This algorithm has been shown to decrease stress monotonically (see de Leeuw[2]).

Use in graph drawing[edit]

Stress majorization and algorithms similar to SMACOF also have application in the field of graph drawing.[4][5] That is, one can find a reasonably aesthetically appealing layout for a network or graph by minimizing a stress function over the positions of the nodes in the graph. In this case, the \delta_{ij} are usually set to the graph-theoretic distances between nodes i and j and the weights w_{ij} are taken to be \delta_{ij}^{-\alpha}. Here, \alpha is chosen as a trade-off between preserving long- or short-range ideal distances. Good results have been shown for \alpha=2.[6]

References[edit]

  1. ^ Kruskal, J. B. (1964), "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis", Psychometrika 29 (1): 1–27, doi:10.1007/BF02289565 .
  2. ^ a b de Leeuw, J. (1977), "Applications of convex analysis to multidimensional scaling", in Barra, J. R.; Brodeau, F.; Romie, G. et al., Recent developments in statistics, pp. 133–145  .
  3. ^ Borg, I.; Groenen, P. (1997), Modern Multidimensional Scaling: theory and applications, New York: Springer-Verlag .
  4. ^ Michailidis, G.; de Leeuw, J. (2001), "Data visualization through graph drawing", Computation Stat. 16 (3): 435–450, doi:10.1007/s001800100077 .
  5. ^ Gansner, E.; Koren, Y.; North, S. (2004), "Graph Drawing by Stress Majorization", Proceedings of 12th Int. Symp. Graph Drawing (GD'04), Lecture Notes in Computer Science 3383, Springer-Verlag, pp. 239–250 .
  6. ^ Cohen, J. (1997), "Drawing graphs to convey proximity: an incremental arrangement method", ACM Transactions on Computer-Human Interaction 4 (3): 197–229, doi:10.1145/264645.264657 .