# Stress majorization

Short description: Geometric placement based on ideal distances

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

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

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

There are many ways that $\displaystyle{ \sigma(X) }$ 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 $\displaystyle{ \sigma }$ from above and touches the surface of $\displaystyle{ \sigma }$ at a point $\displaystyle{ Z }$, 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 COmplicated Function").

## The SMACOF algorithm

The stress function $\displaystyle{ \sigma }$ can be expanded as follows:

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

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

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

where $\displaystyle{ B(Z) }$ has:

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

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

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

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

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

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

The iterative minimization procedure is then:

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

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

## Use in graph drawing

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 $\displaystyle{ \delta_{ij} }$ are usually set to the graph-theoretic distances between nodes $\displaystyle{ i }$ and $\displaystyle{ j }$ and the weights $\displaystyle{ w_{ij} }$ are taken to be $\displaystyle{ \delta_{ij}^{-\alpha} }$. Here, $\displaystyle{ \alpha }$ is chosen as a trade-off between preserving long- or short-range ideal distances. Good results have been shown for $\displaystyle{ \alpha=2 }$.[6]

## References

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. 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 .